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Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.
A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.
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Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.
Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.
The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.
The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.
In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.
All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.
Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:
We will learn more about these test statistics in the upcoming section.
Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.
A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:
The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.
The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.
One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.
Right Tailed Hypothesis Testing
The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:
\(H_{0}\): The population parameter is ≤ some value
\(H_{1}\): The population parameter is > some value.
If the test statistic has a greater value than the critical value then the null hypothesis is rejected
Left Tailed Hypothesis Testing
The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:
\(H_{0}\): The population parameter is ≥ some value
\(H_{1}\): The population parameter is < some value.
The null hypothesis is rejected if the test statistic has a value lesser than the critical value.
In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:
\(H_{0}\): the population parameter = some value
\(H_{1}\): the population parameter ≠ some value
The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.
Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:
The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.
Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.
Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.
Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.
1 - \(\alpha\) = 1 - 0.05 = 0.95
0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.
Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.
z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15
z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56
Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.
Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.
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Important Notes on Hypothesis Testing
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What is hypothesis testing.
Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.
The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.
The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.
The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).
The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.
When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.
To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.
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Priya ranganathan.
1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India
2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India
The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.
Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.
Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).
A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).
The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.
A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.
There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.
The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.
The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.
Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).
Table 1 gives a summary of the two types of statistical errors with an example
Statistical errors
(a) Types of statistical errors | |||
: Null hypothesis is | |||
True | False | ||
Null hypothesis is actually | True | Correct results! | Falsely rejecting null hypothesis - Type I error |
False | Falsely accepting null hypothesis - Type II error | Correct results! | |
(b) Possible statistical errors in the ABLE trial | |||
There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | ||
Truth | There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Correct results! | Falsely rejecting null hypothesis - Type I error |
There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Falsely accepting null hypothesis - Type II error | Correct results! |
In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.
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Conflict of interest: None
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5.2 - writing hypotheses.
The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).
When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.
Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).
Research Question | Is the population mean different from \( \mu_{0} \)? | Is the population mean greater than \(\mu_{0}\)? | Is the population mean less than \(\mu_{0}\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu\neq \mu_{0} \) | \(\mu> \mu_{0} \) | \(\mu<\mu_{0} \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is there a difference in the population? | Is there a mean increase in the population? | Is there a mean decrease in the population? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_d=0 \) | \(\mu_d =0 \) | \(\mu_d=0 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_d \neq 0 \) | \(\mu_d> 0 \) | \(\mu_d<0 \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the population proportion different from \(p_0\)? | Is the population proportion greater than \(p_0\)? | Is the population proportion less than \(p_0\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p=p_0\) | \(p= p_0\) | \(p= p_0\) |
Alternative Hypothesis, \(H_{a}\) | \(p\neq p_0\) | \(p> p_0\) | \(p< p_0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population means different? | Is the population mean in group 1 greater than the population mean in group 2? | Is the population mean in group 1 less than the population mean in groups 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_1=\mu_2\) | \(\mu_1 = \mu_2 \) | \(\mu_1 = \mu_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_1 \ne \mu_2 \) | \(\mu_1 \gt \mu_2 \) | \(\mu_1 \lt \mu_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population proportions different? | Is the population proportion in group 1 greater than the population proportion in groups 2? | Is the population proportion in group 1 less than the population proportion in group 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p_1 = p_2 \) | \(p_1 = p_2 \) | \(p_1 = p_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(p_1 \ne p_2\) | \(p_1 \gt p_2 \) | \(p_1 \lt p_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the slope in the population different from 0? | Is the slope in the population positive? | Is the slope in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\beta =0\) | \(\beta= 0\) | \(\beta = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\beta\neq 0\) | \(\beta> 0\) | \(\beta< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the correlation in the population different from 0? | Is the correlation in the population positive? | Is the correlation in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\rho=0\) | \(\rho= 0\) | \(\rho = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\rho \neq 0\) | \(\rho > 0\) | \(\rho< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
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The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.
When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.
The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.
The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.
The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).
A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).
The level of statistical significance is often expressed as a p-value between 0 and 1.
The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.
Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.
Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.
The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.
A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.
This suggests the effect under study likely represents a real relationship rather than just random chance.
For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05.
It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).
Therefore, we reject the null hypothesis and accept the alternative hypothesis.
Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.
A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.
Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.
Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.
This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.
Note : when the p-value is above your threshold of significance, it does not mean that there is a 95% probability that the alternative hypothesis is true.
Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.
Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.
These tables help you understand how often you would expect to see your test statistic under the null hypothesis.
Understanding the Statistical Test:
Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.
For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.
Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.
This factor is particularly important to consider when comparing results across different analyses.
If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.
A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).
Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).
In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).
The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:
“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.
The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)
A lower p-value is sometimes interpreted as meaning there is a stronger relationship between two variables.
However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).
To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .
In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.
Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.
The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.
If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.
No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.
A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.
Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.
Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.
With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.
Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.
No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.
There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.
Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.
While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001
Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply. BMJ: British Medical Journal , 309 (6958), 874.
Goodman, S. N., & Royall, R. (1988). Evidence and scientific research. American Journal of Public Health , 78 (12), 1568-1574.
Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In Seminars in hematology (Vol. 45, No. 3, pp. 135-140). WB Saunders.
Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value. Epidemiology (Cambridge, Mass.) , 9 (1), 7-8.
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It's the initial building block in the scientific method.
What makes a hypothesis testable.
Bibliography.
A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research.
The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).
A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.
A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .
Here are some examples of hypothesis statements:
A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."
An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.
In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .
For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."
If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (BCcampus, 2015).
There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.
Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley .
A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.
The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.
"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts."
Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis
Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.
California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm
Karl Popper, "Conjectures and Refutations," Routledge, 1963.
Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.
University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf
William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/
University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf
University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19
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Statistics By Jim
Making statistics intuitive
By Jim Frost 10 Comments
A confidence interval (CI) is a range of values that is likely to contain the value of an unknown population parameter . These intervals represent a plausible domain for the parameter given the characteristics of your sample data. Confidence intervals are derived from sample statistics and are calculated using a specified confidence level.
Population parameters are typically unknown because it is usually impossible to measure entire populations. By using a sample, you can estimate these parameters. However, the estimates rarely equal the parameter precisely thanks to random sampling error . Fortunately, inferential statistics procedures can evaluate a sample and incorporate the uncertainty inherent when using samples. Confidence intervals place a margin of error around the point estimate to help us understand how wrong the estimate might be.
You’ll frequently use confidence intervals to bound the sample mean and standard deviation parameters. But you can also create them for regression coefficients , proportions, rates of occurrence (Poisson), and the differences between populations.
Related post : Populations, Parameters, and Samples in Inferential Statistics
The confidence level is the long-run probability that a series of confidence intervals will contain the true value of the population parameter.
Different random samples drawn from the same population are likely to produce slightly different intervals. If you draw many random samples and calculate a confidence interval for each sample, a percentage of them will contain the parameter.
The confidence level is the percentage of the intervals that contain the parameter. For 95% confidence intervals, an average of 19 out of 20 include the population parameter, as shown below.
The image above shows a hypothetical series of 20 confidence intervals from a study that draws multiple random samples from the same population. The horizontal red dashed line is the population parameter, which is usually unknown. Each blue dot is a the sample’s point estimate for the population parameter. Green lines represent CIs that contain the parameter, while the red line is a CI that does not contain it. The graph illustrates how confidence intervals are not perfect but usually correct.
The CI procedure provides meaningful estimates because it produces ranges that usually contain the parameter. Hence, they present plausible values for the parameter.
Technically, you can create CIs using any confidence level between 0 and 100%. However, the most common confidence level is 95%. Analysts occasionally use 99% and 90%.
Related posts : Populations and Samples and Parameters vs. Statistics ,
A confidence interval indicates where the population parameter is likely to reside. For example, a 95% confidence interval of the mean [9 11] suggests you can be 95% confident that the population mean is between 9 and 11.
Confidence intervals also help you navigate the uncertainty of how well a sample estimates a value for an entire population.
These intervals start with the point estimate for the sample and add a margin of error around it. The point estimate is the best guess for the parameter value. The margin of error accounts for the uncertainty involved when using a sample to estimate an entire population.
The width of the confidence interval around the point estimate reveals the precision. If the range is narrow, the margin of error is small, and there is only a tiny range of plausible values. That’s a precise estimate. However, if the interval is wide, the margin of error is large, and the actual parameter value is likely to fall somewhere within that more extensive range . That’s an imprecise estimate.
Ideally, you’d like a narrow confidence interval because you’ll have a much better idea of the actual population value!
For example, imagine we have two different samples with a sample mean of 10. It appears both estimates are the same. Now let’s assess the 95% confidence intervals. One interval is [5 15] while the other is [9 11]. The latter range is narrower, suggesting a more precise estimate.
That’s how CIs provide more information than the point estimate (e.g., sample mean) alone.
Related post : Precision vs. Accuracy
Confidence intervals are similarly helpful for understanding an effect size. For example, if you assess a treatment and control group, the mean difference between these groups is the estimated effect size. A 2-sample t-test can construct a confidence interval for the mean difference.
In this scenario, consider both the size and precision of the estimated effect. Ideally, an estimated effect is both large enough to be meaningful and sufficiently precise for you to trust. CIs allow you to assess both of these considerations! Learn more about this distinction in my post about Practical vs. Statistical Significance .
Learn more about how confidence intervals and hypothesis tests are similar .
Related post : Effect Sizes in Statistics
A frequent misuse is applying confidence intervals to the distribution of sample values. Remember that these ranges apply only to population parameters, not the data values.
For example, a 95% confidence interval [10 15] indicates that we can be 95% confident that the parameter is within that range.
However, it does NOT indicate that 95% of the sample values occur in that range.
If you need to use your sample to find the proportion of data values likely to fall within a range, use a tolerance interval instead.
Related post : See how confidence intervals compare to prediction intervals and tolerance intervals .
Ok, so you want narrower CIs for their greater precision. What conditions produce tighter ranges?
Sample size, variability, and the confidence level affect the widths of confidence intervals. The first two are characteristics of your sample, which I’ll cover first.
Variability present in your data affects the precision of the estimate. Your confidence intervals will be broader when your sample standard deviation is high.
It makes sense when you think about it. When there is a lot of variability present in your sample, you’re going to be less sure about the estimates it produces. After all, a high standard deviation means your sample data are really bouncing around! That’s not conducive for finding precise estimates.
Unfortunately, you often don’t have much control over data variability. You can institute measurement and data collection procedures that reduce outside sources of variability, but after that, you’re at the mercy of the variability inherent in your subject area. But, if you can reduce external sources of variation, that’ll help you reduce the width of your confidence intervals.
Increasing your sample size is the primary way to reduce the widths of confidence intervals because, in most cases, you can control it more than the variability. If you don’t change anything else and only increase the sample size, the ranges tend to narrow. Need even tighter CIs? Just increase the sample size some more!
Theoretically, there is no limit, and you can dramatically increase the sample size to produce remarkably narrow ranges. However, logistics, time, and cost issues will constrain your maximum sample size in the real world.
In summary, larger sample sizes and lower variability reduce the margin of error around the point estimate and create narrower confidence intervals. I’ll point out these factors again when we get to the formula later in this post.
Related post : Sample Statistics Are Always Wrong (to Some Extent)!
The confidence level also affects the confidence interval width. However, this factor is a methodology choice separate from your sample’s characteristics.
If you increase the confidence level (e.g., 95% to 99%) while holding the sample size and variability constant, the confidence interval widens. Conversely, decreasing the confidence level (e.g., 95% to 90%) narrows the range.
I’ve found that many students find the effect of changing the confidence level on the width of the range to be counterintuitive.
Imagine you take your knowledge of a subject area and indicate you’re 95% confident that the correct answer lies between 15 and 20. Then I ask you to give me your confidence for it falling between 17 and 18. The correct answer is less likely to fall within the narrower interval, so your confidence naturally decreases.
Conversely, I ask you about your confidence that it’s between 10 and 30. That’s a much wider range, and the correct value is more likely to be in it. Consequently, your confidence grows.
Confidence levels involve a tradeoff between confidence and the interval’s spread. To have more confidence that the parameter falls within the interval, you must widen the interval. Conversely, your confidence necessarily decreases if you use a narrower range.
Confidence intervals account for sampling uncertainty by using critical values, sampling distributions, and standard errors. The precise formula depends on the type of parameter you’re evaluating. The most common type is for the mean, so I’ll stick with that.
You’ll use critical Z-values or t-values to calculate your confidence interval of the mean. T-values produce more accurate confidence intervals when you do not know the population standard deviation. That’s particularly true for sample sizes smaller than 30. For larger samples, the two methods produce similar results. In practice, you’d usually use a t-value.
Below are the confidence interval formulas for both Z and t. However, you’d only use one of them.
The only difference between the two formulas is the critical value. If you’re using the critical z-value, you’ll always use 1.96 for 95% confidence intervals. However, for the t-value, you’ll need to know the degrees of freedom and then look up the critical value in a t-table or online calculator.
To calculate a confidence interval, take the critical value (Z or t) and multiply it by the standard error of the mean (SEM). This value is known as the margin of error (MOE) . Then add and subtract the MOE from the sample mean (x̄) to produce the upper and lower limits of the range.
Related posts : Critical Values , Standard Error of the Mean , and Sampling Distributions
Think back to the discussion about the factors affecting the confidence interval widths. The formula helps you understand how that works. Recall that the critical value * SEM = MOE.
Smaller margins of error produce narrower confidence intervals. By looking at this equation, you can see that the following conditions create a smaller MOE:
Let’s move on to using these formulas to find a confidence interval! For this example, I’ll use a fuel cost dataset that I’ve used in other posts: FuelCosts . The dataset contains a random sample of 25 fuel costs. We want to calculate the 95% confidence interval of the mean.
However, imagine we have only the following summary information instead of the dataset.
Fortunately, that’s all we need to calculate our 95% confidence interval of the mean.
We need to decide on using the critical Z or t-value. I’ll use a critical t-value because the sample size (25) is less than 30. However, if the summary didn’t provide the sample size, we could use the Z-value method for an approximation.
My next step is to look up the critical t-value using my t-table. In the table, I’ll choose the alpha that equals 1 – the confidence level (1 – 0.95 = 0.05) for a two-sided test. Below is a truncated version of the t-table. Click for the full t-distribution table .
In the table, I see that for a two-sided interval with 25 – 1 = 24 degrees of freedom and an alpha of 0.05, the critical value is 2.064.
Let’s enter all of this information into the formula.
First, I’ll calculate the margin of error:
Next, I’ll take the sample mean and add and subtract the margin of error from it:
The 95% confidence interval of the mean for fuel costs is 267.0 – 394.2. We can be 95% confident that the population mean falls within this range.
If you had used the critical z-value (1.96), you would enter that into the formula instead of the t-value (2.064) and obtain a slightly different confidence interval. However, t-values produce more accurate results, particularly for smaller samples like this one.
As an aside, the Z-value method always produces narrower confidence intervals than t-values when your sample size is less than infinity. So, basically always! However, that’s not good because Z-values underestimate the uncertainty when you’re using a sample estimate of the standard deviation rather than the actual population value. And you practically never know the population standard deviation.
Neyman, J. (1937). Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability . Philosophical Transactions of the Royal Society A . 236 (767): 333–380.
April 23, 2024 at 8:37 am
February 24, 2024 at 8:29 am
Thank you so much
February 14, 2024 at 1:56 pm
If I take a sample and create a confidence interval for the mean, can I say that 95% of the mean of the other samples I will take can be found in this range?
February 23, 2024 at 8:40 pm
Unfortunately, that would be an invalid statement. The CI formula uses your sample to estimate the properties of the population to construct the CI. Your estimates are bound to be off by at least a little bit. If you knew the precise properties of the population, you could determine the range in which 95% of random samples from that population would fall. However, again, you don’t know the precise properties of the population. You just have estimates based on your sample.
September 29, 2023 at 6:55 pm
Hi Jim, My confusion is similar to one comment. What I cannot seem to understand is the concept of individual and many CIs and therefore statements such as X% of the CIs.
For a sampling distribution, which itself requires many samples to produce, we try to find a confidence interval. Then how come there are multiple CIs. More specifically “Different random samples drawn from the same population are likely to produce slightly different intervals. If you draw many random samples and calculate a confidence interval for each sample, a percentage of them will contain the parameter.” this is what confuses me. Is interval here represents the range of the samples drawn? If that is true, why is the term CI or interval used for sample range? If not, could you please explain what is mean by an individual CI or how are we calculating confidence interval for each sample? In the image depicting 19 out of 20 will have population parameter, is the green line the range of individual samples or the confidence interval?
Please try to sort this confusion out for me. I find your website really helpful for clearing my statistical concepts. Thank you in advance for helping out. Regards.
September 30, 2023 at 1:52 am
A key point to remember is that inferential statistics occur in the context of drawing many random samples from the same population. Of course, a single study typically draws a single sample. However, if that study were to draw another random sample, it would be somewhat different than the first sample. A third sample would be somewhat different as well. That produces the sampling distribution, which helps you calculate p-values and construct CIs. Inferential statistics procedures use the idea of many samples to incorporate random sampling error into the results.
For CIs, if you were to collect many random samples, a certain percentage of them will contain the population parameter. That percentage is the confidence interval. Again, a single study will only collect a single sample. However, picturing many CIs helps you understand the concept of the confidence level. In practice, a study generates one CI per parameter estimate. But the graph with multiple CIs is just to help you understand the concept of confidence level.
Alternatively, you can think of CIs as an object class. Suppose 100 disparate studies produce 95% CIs. You can assume that about 95 of those CIs actually contain the population parameter. Using statistical procedures, you can estimate the sampling distribution using the sample itself without collecting many samples.
I don’t know what you mean by “Interval here represents the range of samples drawn.” As I write in this article, the CI is an interval of values that likely contain the population parameter. Reread the section titled How to Interpret Confidence Intervals to understand what each one means.
Each CI is estimated from a single sample and a study generates one CI per parameter estimate. However, again, understanding the concept of the confidence level is easier when you picture multiple CIs. But if a single study were to collect multiple samples and produces multiple CIs, that graph is what you’d expect to see. Although, in the real world, you never know for sure whether a CI actually contains the parameter or not.
The green lines represent CIs that contain the population parameter. Red lines represent CIs that do not contain the population parameter. The graph illustrates how CIs are not perfect but they are usually correct. I’ve added text to the article to clarify that image.
I also show you how to calculate the CI for a mean in this article. I’m not sure what more you need to understand there? I’m happy to clarify any part of that.
I hope that helps!
July 6, 2023 at 10:14 am
Hi Jim, This was an excellent article, thank you! I have a question: when computing a CI in its single-sample t-test module, SPSS appears to use the difference between population and sample means as a starting point (so the formula would be (X-bar-mu) +/- tcv(SEM)). I’ve consulted multiple stats books, but none of them compute a CI that way for a single-sample t-test. Maybe I’m just missing something and this is a perfectly acceptable way of doing things (I mean, SPSS does it :-)), but it yields substantially different lower and upper bounds from a CI that uses the traditional X-bar as a starting point. Do you have any insights? Many thanks in advance! Stephen
July 7, 2023 at 2:56 am
Hi Stephen,
I’m not an SPSS user but that formula is confusing. They presented this formula as being for the CI of a sample mean?
I’m not sure why they’re subtracting Mu. For one thing, you almost never know what Mu is because you’d have to measure the entire population. And, if you knew Mu, you wouldn’t need to perform a t-test! Why would you use a sample mean (X-bar) if you knew the population mean? None of it makes sense to me. It must be an error of some kind even if just of documentation.
October 13, 2022 at 8:33 am
Are there strict distinctions between the terms “confident”, “likely”, and “probability”? I’ve seen a number of other sources exclaim that for a given calculated confidence interval, the frequentist interpretation of that is the parameter is either in or not in that interval. They say another frequent misinterpretation is that the parameter lies within a calculated interval with a 95% probability.
It’s very confusing to balance that notion with practical casual communication of data in non-research settings.
October 13, 2022 at 5:43 pm
It is a confusing issue.
In this strictest technical sense, the confidence level is probability that applies to the process but NOT an individual confidence interval. There are several reasons for that.
In the frequentist framework, the probability that an individual CI contains the parameter is either 100% or 0%. It’s either in it or out. The parameter is not a random variable. However, because you don’t know the parameter value, you don’t know which of those two conditions is correct. That’s the conceptual approach. And the mathematics behind the scenes are complementary to that. There’s just no way to calculate the probability that an individual CI contains the parameter.
On the other hand, the process behind creating the intervals will cause X% of the CIs at the Xth confidence level to include that parameter. So, for all 95% CIs, you’d expect 95% of them to contain the parameter value. The confidence level applies to the process, not the individual CIs. Statisticians intentionally used the term “confidence” to describe that as opposed to “probability” hoping to make that distinction.
So, the 95% confidence applies the process but not individual CIs.
However, if you’re thinking that if 95% of many CIs contain the parameter, then surely a single CI has a 95% probability. From a technical standpoint, that is NOT true. However, it sure sounds logical. Most statistics make intuitive sense to me, but I struggle with that one myself. I’ve asked other statisticians to get their take on it. The basic gist of their answers is that there might be other information available which can alter the actual probability. Not all CIs produced by the process have the same probability. For example, if an individual CI is a bit higher or lower than most other CIs for the same thing, the CIs with the unusual values will have lower probabilities for containing the parameters.
I think that makes sense. The only problem is that you often don’t know where your individual CI fits in. That means you don’t know the probability for it specifically. But you do know the overall probability for the process.
The answer for this question is never totally satisfying. Just remember that there is no mathematical way in the frequentist framework to calculate the probability that an individual CI contains the parameter. However, the overall process is designed such that all CIs using a particular confidence level will have the specified proportion containing the parameter. However, you can’t apply that overall proportion to your individual CI because on the technical side there’s no mathematical way to do that and conceptually, you don’t know where your individual CI fits in the entire distribution of CIs.
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Hypotheses are predictions of expected findings.
A research hypothesis is a mathematical way of stating a research question. A research hypothesis names the groups (we'll start with a sample and a population), what was measured, and which we think will have a higher mean. The last one gives the research hypothesis a direction. In other words, a research hypothesis should include:
There are two types of research hypotheses related to sample means and population means: Directional Research Hypotheses and Non-Directional Research Hypotheses
If we expect our obtained sample mean to be above or below the other group's mean (the population mean, for example), we have a directional hypothesis. There are two options:
A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework. What’s a directional hypothesis for how scoring higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework? Write this out in words and symbols.
Answer in Words: Students who scored high on growth mindset would spend more time on their homework than the population of junior high students.
Answer in Symbols: \( \displaystyle \bar{X} > \mu \)
A non-directional hypothesis states that the means will be different, but does not specify which will be higher. In reality, there is rarely a situation in which we actually don't want one group to be higher than the other, so we will focus on directional research hypotheses. There is only one option for a non-directional research hypothesis: "The sample mean differs from the population mean." These types of research hypotheses don’t give a direction, the hypothesis doesn’t say which will be higher or lower.
A non-directional research hypothesis in symbols should look like this: \( \displaystyle \bar{X} \neq \mu \) (The mean of the sample is not equal to the mean of the population).
What’s a non-directional hypothesis for how scoring higher on growth mindset higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework (Blackwell, Trzesniewski, & Dweck, 2007)? Write this out in words and symbols.
Answer in Words: Students who scored high on growth mindset would spend a different amount of time on their homework than the population of junior high students.
Answer in Symbols: \( \displaystyle \bar{X} \neq \mu \)
See how a non-directional research hypothesis doesn't really make sense? The big issue is not if the two groups differ, but if one group seems to improve what was measured (if having a growth mindset leads to more time spent on math homework). This textbook will only use directional research hypotheses because researchers almost always have a predicted direction (meaning that we almost always know which group we think will score higher).
The hypothesis that an apparent effect is due to chance is called the null hypothesis, written \(H_0\) (“H-naught”). We usually test this through comparing an experimental group to a comparison (control) group. This null hypothesis can be written as:
\[\mathrm{H}_{0}: \bar{X} = \mu \nonumber \]
For most of this textbook, the null hypothesis is that the means of the two groups are similar. Much later, the null hypothesis will be that there is no relationship between the two groups. Either way, remember that a null hypothesis is always saying that nothing is different.
This is where descriptive statistics diverge from inferential statistics. We know what the value of \(\overline{\mathrm{X}}\) is – it’s not a mystery or a question, it is what we observed from the sample. What we are using inferential statistics to do is infer whether this sample's descriptive statistics probably represents the population's descriptive statistics. This is the null hypothesis, that the two groups are similar.
Keep in mind that the null hypothesis is typically the opposite of the research hypothesis. A research hypothesis for the ESP example is that those in my sample who say that they have ESP would get more correct answers than the population would get correct, while the null hypothesis is that the average number correct for the two groups will be similar.
In general, the null hypothesis is the idea that nothing is going on: there is no effect of our treatment, no relation between our variables, and no difference in our sample mean from what we expected about the population mean. This is always our baseline starting assumption, and it is what we seek to reject. If we are trying to treat depression, we want to find a difference in average symptoms between our treatment and control groups. If we are trying to predict job performance, we want to find a relation between conscientiousness and evaluation scores. However, until we have evidence against it, we must use the null hypothesis as our starting point.
In sum, the null hypothesis is always : There is no difference between the groups’ means OR There is no relationship between the variables .
In the next chapter, the null hypothesis is that there’s no difference between the sample mean and population mean. In other words:
A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework. What’s the null hypothesis for scoring higher on growth mindset (compared to the population of junior high students) and how long students spent on their homework? Write this out in words and symbols.
Answer in Words: Students who scored high on growth mindset would spend a similar amount of time on their homework as the population of junior high students.
Answer in Symbols: \( \bar{X} = \mu \)
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Published on September 4, 2020 by Pritha Bhandari . Revised on June 22, 2023.
While descriptive statistics summarize the characteristics of a data set, inferential statistics help you come to conclusions and make predictions based on your data.
When you have collected data from a sample , you can use inferential statistics to understand the larger population from which the sample is taken.
Inferential statistics have two main uses:
Descriptive versus inferential statistics, estimating population parameters from sample statistics, hypothesis testing, other interesting articles, frequently asked questions about inferential statistics.
Descriptive statistics allow you to describe a data set, while inferential statistics allow you to make inferences based on a data set.
Using descriptive statistics, you can report characteristics of your data:
In descriptive statistics, there is no uncertainty – the statistics precisely describe the data that you collected. If you collect data from an entire population, you can directly compare these descriptive statistics to those from other populations.
Most of the time, you can only acquire data from samples, because it is too difficult or expensive to collect data from the whole population that you’re interested in.
While descriptive statistics can only summarize a sample’s characteristics, inferential statistics use your sample to make reasonable guesses about the larger population.
With inferential statistics, it’s important to use random and unbiased sampling methods . If your sample isn’t representative of your population, then you can’t make valid statistical inferences or generalize .
Since the size of a sample is always smaller than the size of the population, some of the population isn’t captured by sample data. This creates sampling error , which is the difference between the true population values (called parameters) and the measured sample values (called statistics).
Sampling error arises any time you use a sample, even if your sample is random and unbiased. For this reason, there is always some uncertainty in inferential statistics. However, using probability sampling methods reduces this uncertainty.
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The characteristics of samples and populations are described by numbers called statistics and parameters :
Sampling error is the difference between a parameter and a corresponding statistic. Since in most cases you don’t know the real population parameter, you can use inferential statistics to estimate these parameters in a way that takes sampling error into account.
There are two important types of estimates you can make about the population: point estimates and interval estimates .
Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.
A confidence interval uses the variability around a statistic to come up with an interval estimate for a parameter. Confidence intervals are useful for estimating parameters because they take sampling error into account.
While a point estimate gives you a precise value for the parameter you are interested in, a confidence interval tells you the uncertainty of the point estimate. They are best used in combination with each other.
Each confidence interval is associated with a confidence level. A confidence level tells you the probability (in percentage) of the interval containing the parameter estimate if you repeat the study again.
A 95% confidence interval means that if you repeat your study with a new sample in exactly the same way 100 times, you can expect your estimate to lie within the specified range of values 95 times.
Although you can say that your estimate will lie within the interval a certain percentage of the time, you cannot say for sure that the actual population parameter will. That’s because you can’t know the true value of the population parameter without collecting data from the full population.
However, with random sampling and a suitable sample size, you can reasonably expect your confidence interval to contain the parameter a certain percentage of the time.
Your point estimate of the population mean paid vacation days is the sample mean of 19 paid vacation days.
Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples.
Hypotheses , or predictions, are tested using statistical tests . Statistical tests also estimate sampling errors so that valid inferences can be made.
Statistical tests can be parametric or non-parametric. Parametric tests are considered more statistically powerful because they are more likely to detect an effect if one exists.
Parametric tests make assumptions that include the following:
When your data violates any of these assumptions, non-parametric tests are more suitable. Non-parametric tests are called “distribution-free tests” because they don’t assume anything about the distribution of the population data.
Statistical tests come in three forms: tests of comparison, correlation or regression.
Comparison tests assess whether there are differences in means, medians or rankings of scores of two or more groups.
To decide which test suits your aim, consider whether your data meets the conditions necessary for parametric tests, the number of samples, and the levels of measurement of your variables.
Means can only be found for interval or ratio data , while medians and rankings are more appropriate measures for ordinal data .
test | Yes | Means | 2 samples |
---|---|---|---|
Yes | Means | 3+ samples | |
Mood’s median | No | Medians | 2+ samples |
Wilcoxon signed-rank | No | Distributions | 2 samples |
Wilcoxon rank-sum (Mann-Whitney ) | No | Sums of rankings | 2 samples |
Kruskal-Wallis | No | Mean rankings | 3+ samples |
Correlation tests determine the extent to which two variables are associated.
Although Pearson’s r is the most statistically powerful test, Spearman’s r is appropriate for interval and ratio variables when the data doesn’t follow a normal distribution.
The chi square test of independence is the only test that can be used with nominal variables.
Pearson’s | Yes | Interval/ratio variables |
---|---|---|
Spearman’s | No | Ordinal/interval/ratio variables |
Chi square test of independence | No | Nominal/ordinal variables |
Regression tests demonstrate whether changes in predictor variables cause changes in an outcome variable. You can decide which regression test to use based on the number and types of variables you have as predictors and outcomes.
Most of the commonly used regression tests are parametric. If your data is not normally distributed, you can perform data transformations.
Data transformations help you make your data normally distributed using mathematical operations, like taking the square root of each value.
1 interval/ratio variable | 1 interval/ratio variable | |
2+ interval/ratio variable(s) | 1 interval/ratio variable | |
Logistic regression | 1+ any variable(s) | 1 binary variable |
Nominal regression | 1+ any variable(s) | 1 nominal variable |
Ordinal regression | 1+ any variable(s) | 1 ordinal variable |
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.
A statistic refers to measures about the sample , while a parameter refers to measures about the population .
A sampling error is the difference between a population parameter and a sample statistic .
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
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Bhandari, P. (2023, June 22). Inferential Statistics | An Easy Introduction & Examples. Scribbr. Retrieved June 9, 2024, from https://www.scribbr.com/statistics/inferential-statistics/
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dc.contributor.advisor | Hofmann, Heike | |
dc.contributor.advisor | Qiu, Yumou | |
dc.contributor.advisor | Yu, Cindy | |
dc.contributor.advisor | Nordman, Daniel | |
dc.contributor.advisor | Liu, Peng | Ge, Yawei | Statistics | en_US | 2024-06-05T22:05:56Z | 2024-06-05T22:05:56Z | 2024-05 | 2024-06-05T22:05:56Z | We consider the applications of statistical methods in the field of forensic science and genetic science, where we are able to record more and better traits from the study objects as the advance of modern technology. In forensic science, we are moving from the traditional microscope comparisons to 3D scans with computer based algorithms in response to the need of objective methods. We propose beta-mixture distributions to model the cross-correlation functions (CCF) for matching and non-matching bullets, and further apply the score-based likelihood ratios to quantify the strength of evidence. We also propose a screening procedure and analyze the use of reference database which improves the performance of prediction. In genetic science, we analyze the experiment data which is designed to understand the underlying relations among biomass, metabolites, microbiome, genotypes of sorghum and levels of nitrogen. The recorded number of mricrobiome and metabolites by advanced measurement technology is larger than the sample size, which is known as the ``large $p$, small $n$" scenario. We propose a high-dimensional linear mixed effect model to quantify the effect of multiple groups of factors to biomass. Further, we propose a high-dimensional test procedure to conduct the simultaneous test of coefficients while controlling a group of high-dimensional nuisance variables to replace the conventional F-test. We show the properties of the proposed test statistic through theorems and simulations. | 0009-0001-8353-1477 | https://dr.lib.iastate.edu/handle/20.500.12876/KrZJLnbr | en | en | Statistics | en_US | forensic | en_US |
dc.subject.keywords | genetic | en_US |
dc.subject.keywords | high-dimensional | en_US |
dc.subject.keywords | hypothesis test | en_US | High-dimensional statistical models and hypothesis tests with a focus on forensic and genetic applications | article | en_US | dissertation | en_US | Publication | Statistics | en_US | Iowa State University | en_US | dissertation | $ | Doctor of Philosophy | en_US |
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Step 5: Present your findings. The results of hypothesis testing will be presented in the results and discussion sections of your research paper, dissertation or thesis.. In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p-value).
10.1 - Setting the Hypotheses: Examples. A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or ...
If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
A statistical hypothesis is an assumption about a population parameter.. For example, we may assume that the mean height of a male in the U.S. is 70 inches. The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter.. A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical ...
The Four Steps in Hypothesis Testing. STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha. STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. If the conditions are met, summarize the data using a test statistic.
A statistical hypothesis test may return a value called p or the p-value. This is a quantity that we can use to interpret or quantify the result of the test and either reject or fail to reject the null hypothesis. This is done by comparing the p-value to a threshold value chosen beforehand called the significance level.
Step 5: Phrase your hypothesis in three ways. To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable. If a first-year student starts attending more lectures, then their exam scores will improve.
In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true. To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect. Step 1: State the null hypothesis: H 0 :μ=100. Step 2: State the alternate hypothesis: H 1 :≠100. Step 3: State your alpha level. We'll use 0.05 for this example.
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Step 1: State the Null Hypothesis. The null hypothesis can be thought of as the opposite of the "guess" the researchers made: in this example, the biologist thinks the plant height will be different for the fertilizers. So the null would be that there will be no difference among the groups of plants. Specifically, in more statistical language ...
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid. A null hypothesis and an alternative ...
HYPOTHESIS TESTING. A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the "alternate" hypothesis, and the opposite ...
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...
Hypothesis testing involves two statistical hypotheses. The first is the null hypothesis (H 0) as described above.For each H 0, there is an alternative hypothesis (H a) that will be favored if the null hypothesis is found to be statistically not viable.The H a can be either nondirectional or directional, as dictated by the research hypothesis. For example, if a researcher only believes the new ...
Statistical inference is the process of using a sample to infer the properties of a population. Statistical procedures use sample data to estimate the characteristics of the whole population from which the sample was drawn. Scientists typically want to learn about a population. When studying a phenomenon, such as the effects of a new medication ...
Figure 7.1.1. Before calculating the probability, it is useful to see how many standard deviations away from the mean the sample mean is. Using the formula for the z-score from chapter 6, you find. z = ¯ x − μo σ / √n = 490 − 500 25 / √30 = − 2.19. This sample mean is more than two standard deviations away from the mean.
A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true). The level of statistical significance is often expressed as a p-value between 0 and 1. The smaller the p -value, the less likely the results occurred by random chance, and the ...
What does a statistical test do? Statistical tests work by calculating a test statistic - a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.. It then calculates a p value (probability value). The p-value estimates how likely it is that you would see the difference described by the test statistic if the null ...
Bibliography. A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method. Many describe it as an ...
A confidence interval (CI) is a range of values that is likely to contain the value of an unknown population parameter. These intervals represent a plausible domain for the parameter given the characteristics of your sample data. Confidence intervals are derived from sample statistics and are calculated using a specified confidence level.
The Research Hypothesis. A research hypothesis is a mathematical way of stating a research question. A research hypothesis names the groups (we'll start with a sample and a population), what was measured, and which we think will have a higher mean. The last one gives the research hypothesis a direction. In other words, a research hypothesis ...
Example: Inferential statistics. You randomly select a sample of 11th graders in your state and collect data on their SAT scores and other characteristics. You can use inferential statistics to make estimates and test hypotheses about the whole population of 11th graders in the state based on your sample data.
We consider the applications of statistical methods in the field of forensic science and genetic science, where we are able to record more and better traits from the study objects as the advance of modern technology. In forensic science, we are moving from the traditional microscope comparisons to 3D scans with computer based algorithms in response to the need of objective methods.