hypothesis testing math ai sl

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• The statement is either about a population parameter or the distribution of the population
• The hypothesis test will look at the probability of observed outcomes happening under set conditions
• The probability found will be compared against a given significance level to determine whether there is evidence to support the statement being made

What are the key terms used in statistical hypothesis testing?

• Every hypothesis test must begin with a clear null hypothesis (what we believe to already be true) and alternative hypothesis (how we believe the data pattern or probability distribution might have changed)
• Such as the mean value of the population
• A one-tailed test is used for testing the distribution or testing whether the parameter has increased (or decreased)
• A two-tailed test is used for testing whether the parameter has changed (either increased or decreased)
• When a null hypothesis is accepted or rejected a statistical inference is made
• Any probability smaller than the significance level would suggest that the event is unlikely to have happened by chance
• The significance level must be set before the hypothesis test is carried out
• The significance level will usually be 1%, 5% or 10%, however it may vary

Conclusions of Hypothesis Testing

How do i decide whether to reject or accept the null hypothesis.

• Your GDC will calculate the test statistic for you
• To decide whether or not to reject the null hypothesis you first need either the p -value or the critical region
• Your GDC will give you the p -value
• If the p -value is less than the significance level then the null hypothesis would be rejected
• If the test statistic falls within the critical region then the null hypothesis would be rejected
• It is the least extreme value that would lead to the rejection of the null hypothesis
• In your exam you will be given the critical value if it is needed

How should a conclusion be written for a hypothesis test?

• Your conclusion must be written in the context of the question
• If rejecting the null hypothesis your conclusion should state that there is sufficient evidence to suggest the null hypothesis is unlikely true
• If accepting the null hypothesis your conclusion should state that there is not enough evidence to suggest null hypothesis is unlikely true
• There is a chance that the test has led to an incorrect conclusion
• a different sample might lead to a different outcome
• You should not state whether this change is an increase or decrease
• You are simply saying there is not enough evidence to reject it

Author: Daniel

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Hypothesis testing.

Hypotheses (Ho/H1), Significance Levels, P-Values, x^2 Tests for Independence & Fit, T-Tests, 1 & 2 Tailed Tests...

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IB Maths AI – Chapter 4 – Statistics and Probability – Correlation Coefficients + Hypothesis Testing (using the amazing and life-changing GDC)

• March 11, 2024
• Matthew Weng

Table of Contents

Introduction, pearson’s r correlation coefficient, spearman’s rank correlation coefficient, common attributes, chi-squared test for independence, chi-squared goodness of fit (gof) test, two sample t-test.

Chapter 4 in the IB Maths AI curriculum is one of the topics that I noticed many people struggled with. I could see why, considering all of its many different rules and GDC algorithms. Fortunately, I think that I am quite OK with this topic, so I am going to detail the rules of the different formulae and tests, and how to find the answers. I will be starting with Correlation Coefficients and Hypothesis Testing.

Here is a link to the IB Maths AI Formula Booklet: https://www.rcboe.org/cms/lib/GA01903614/Centricity/Domain/1351/Formula%20Book.pdf

Correlation Coefficients

The Pearson Correlation Coefficient is used to find the strength of the correlation between 2 variables . For example, finding the correlation strength between 2 people and their reaction times.

The r-value always lies in the range:

• r = 1, perfect positive linear correlation
• r = -1, perfect negative linear correlation
• r = 0, no linear correlation

Essentially, the closer to 1, the stronger the correlation, and the closer to 0, the weaker the correlation.

Now, how to find the r-value? You could calculate manually, but your life would be made so much easier with a Graphing Digital Calculator (GDC). In this case, I will be using the Ti-nspire CX II.

Here are the steps to finding the r-value:

• Open up a spreadsheet and enter the values provided in the question. Don’t forget to label each column. (Figure 1)
• Once done, put the cursor on an empty column and conduct a Linear Regression (mx+b) test. To do this, click MENU – Statistics – Stat Calculations – Linear Regression (mx+b). SHORTCUT: [MENU – 4 – 1 – 3]
• From here, input the label of your columns into the X and Y list boxes. There is no need to input anything else in other boxes. (Figure 2)
• The results will be outputted, and the r-value is given. (Figure 3)

Sometimes, the question will as you to plot in a linear regression graph/diagram. Here is how that would look like:

• Go to the ‘Data and Statistics’ feature of the GDC.
• Take the X and Y list from before and input it into the X and Y axes of the graph. (Figure 4)
• Display the linear regression line. To do this, click MENU – Regression – Show Linear (mx+b) SHORTCUT: [MENU – 4 – 6 – 1] (Figure 5)

The Spearman’s Coefficient is relatively similar to the Pearson’s Coefficient, but before the r-value can be found, the data in the columns have to be ranked first. In terms of the r-value, it also follows the same rules as the Pearson’s Coefficient.

To find the r s value, the data will be ranked in descending order (The largest value will be rank 1).

Let’s follow a worked example:

From here, a Linear Regression (mx + b) test can be conducted to find the r s value. This is done in the same fashion as the Pearson’s test, just with the ranked data. I will put the shortcut here as a reminder: [MENU – 4 – 1 – 3]

In this question, the r s value is:

r s = 0.359

Hypothesis Testing

Before getting into the different hypothesis tests, I would like to mention that all of these methods use the common value, p . This p-value will be used to determine the outcome of the test. Additionally, all of these hypothesis tests involve a Significance Level. In most cases, the p-value will be compared with the significance level to determine the outcome of the test. Most of the time, the question will provide a significance level, but if it not, a commonly observed significance level is 5%, or 0.05 in decimal.

The Chi-Squared Independence Test is used to suggest an association between 2 variables, and whether these 2 variables are dependent or independent of each other. For example, an example question would be most people who study physics also study higher-level mathematics.

The Chi-Squared Independence Test follows this rule:

p-value < significance level, reject null hypothesis

Following this, here is a rough idea how to carry out this test:

• Identify the null and alternative hypotheses, H 0 and H 1 .
• Open the Calculator function of the GDC.
• Plot the raw data from the question into a matrix, EXCLUDING THE TOTAL VALUES Menu – Matrix & Vector – Create – Matrix
• Initialise the matrix by labelling it ‘observed’ [ctrl – var – type observed]
• Conduct a Chi-Squared Test for Independence. To do this, click Menu – Statistics – Stat Tests – X 2 2-way Test

Let’s follow through a sample question to understand how this hypothesis test works.

In most occasions, the question would provide the data in a table:

From here, we can determine the null and alternative hypotheses of this test:

H 0 = The customer age and chosen meal is independent

H 1 = The customer age and chosen meal is not independent/dependent

From here, we can add the data into a matrix. SHORTCUT: [MENU – 7 – 1 – 1] It also been initialised as ‘obverved’ [ctrl – var – type observed].

From here, the test can be done. SHORTCUT: [MENU – 6 – 7 – 8]. Choose the observed matrix, and click enter:

Conclusion: Since p-value > significance level, 0.264 > 0.100, we have enough evidence to accept the null hypothesis, meaning the customer age and their chosen meal is independent.

The Chi-Squared GOF Test is used to determine whether data fits a particular distribution.

Here is a rough idea on how to conduct a Chi-Squared GOF test:

• Input data into the Spreadsheet function. There will be 2 columns: Observed and Expected
• The observed values will be provided in the question. Sometimes, the expected values will not be provided. This can be calculated by totalling the observed values, and dividing by the number of columns.
• Determine the degrees of freedom (df) [number of columns – 1 = df]
• Conduct a Chi-Squared GOF test. To do this, click Menu – Statistics – Stat Tests – X 2 GOF Test.
• Compare the p-value with the significance level.

Let’s follow through an example to better understand how to conduct a GOF test.

The data will be arranged in a table, similar to the other Hypothesis Tests:

Firstly, let’s determine the null and alternate hypotheses:

H 0 = The same number of copies will be sold each day.

H 1 = A different number of copies will be sold each day

Alternatively, you can express this in numerals:

H 0 = P 1 = 20%, P 2 = 20%, P 3 = 20%, …

H 1 = P 1 not equal to 20%, …

Then, we can determine the Degrees of Freedom (df):

df = 5 – 1 = 4

From here, we can input the data into a Spreadsheet. The expected values will be:

74 + 97 + 91 + 86 + 112 = 460

460 / 5 = 92

Here is what that looks like:

From here, we can conduct the Chi-Squared GOF Test. SHORTCUT: [MENU – 4 – 4 – 7]. Input the observed list, expected list, and degrees of freedom:

From here, we can make a conclusion:

Since p-value > significance level, 0.07 > 0.05, we have enough evidence to accept the null hypothesis, meaning the same number of newspapers will be sold each day.

The Two-Sample t-test is used when comparing 2 means of 2 sets of data. There are other types of t-tests, but most IB questions use the Two-Sample t-test, so I will be focusing on that.

Here is a brief rundown on how to conduct a Two-Sample t-test:

• Input the given data into the Spreadsheet function.
• Conduct a Two-Sample t-test. To do this click Menu – Statistics – Stat Tests – 2-Sample t Test.
• Input the 1st and 2nd lists, change the Alternate Hypothesis to what the question dictates, and click ‘yes’ for the pooled column.
• Compare the p-value and significance level.

Let’s follow through a sample question to better understand how the Two-Sample t-test works:

H 0 = Chinchilla rabbits have the same weight as sable rabbits

H 1 = Chinchilla rabbits have a greater weight than sable rabbits

Then, we can input the data in a spreadsheet:

From here, we can conduct the Two-Sample t-test. SHORTCUT: [MENU – 4 – 4 – 4]. In the case of the question, the alternate hypothesis will be U 0 > U 1 . The results are as follows:

Since p-value < significance level, 0.04 < 0.05, we have enough evidence to reject the null hypothesis, meaning that the weight of chinchilla rabbits is greater than sable rabbits.

I hope this small guide to the correlation coefficients and hypothesis testing will help you, whether you are reviewing or are re-learning the topics. I used actual IB sample questions to better angle my blog towards us IB students, so I hope that helps. Anyways, thank you for reading!

For other review material for other IB subjects, please check out Prodat Blog, where we have review material for Economics, Business Management, and Psychology: https://prodatblog.org/ib-revision-guides/

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The t-test ( DP IB Maths: AI SL )

Revision note.

Two-Sample Tests

What is a t -test.

• A t -test is used to compare the means of two normally distributed populations
• In the exam the population variance will always be unknown

What assumptions are needed for the t -test?

• The underlying distribution for each variable must be normal
• You will need to use the pooled two-sample t -test

What are the steps for a pooled two-sample t -test?

• Where μ x and μ y are the population means
• Make sure you make it clear which mean corresponds to each population
• In words this means the two population means are equal
• The alternative hypothesis will depend on what is being tested (see sections for one-tailed and two-tailed tests)
• Enter two lists of data – one for each sample
• Choose the pooled option
• Your GDC will then give you the p -value
• If p -value < significance level then reject H 0
• If p -value > significance level then accept H 0
• There is sufficient evidence to suggest that the population mean of X is bigger than/smaller than/different to the population mean of Y
• This will depend on the alternative hypothesis
• There is insufficient evidence to suggest that the population mean of X is bigger than/small than/different to the population mean of Y
• Therefore this suggests that the population means are equal

One-tailed Tests

How do i perform a one-tailed t -test.

• The alternative hypothesis will be: H 1 : μ x > μ y
• Look out for words such as increase, bigger, higher, etc
• The alternative hypothesis will be: H 1 : μ x < μ y
• Look out for words such as decrease, smaller, lower, etc
• If the alternative hypothesis is H 1 : μ x > μ y
• If the alternative hypothesis is H 1 : μ x < μ y

Worked example

The times (in minutes) for children and adults to complete a puzzle are recorded below.

The creator of the puzzle claims children are generally faster at solving the puzzle than adults. A t -test is to be performed at a 1% significance level.

Two-tailed Tests

How do i perform a two-tailed t -test.

• The alternative hypothesis will be: H 1 : μ x ≠ μ y
• Look out for words such as change, different, not the same, etc
• This suggests that the population mean of X is different to the population mean of Y
• All you can conclude is that there is evidence that the means are not equal
• To test whether a specific one is bigger you would need to use a one-tailed test

In a school all students must study either French or Spanish as well as maths. 18 students in a maths class complete a test and their scores are recorded along with which language they study.

The maths teacher wants to investigate whether the scores are different between the students studying each language. A t -test is to be performed at a 10% significance level.

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Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.