The t-test (DP IB Applications & Interpretation (AI)): Revision Note
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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
In the exam you will need to assume the variance for the two groups are equal
You will need to use the pooled two-sample t-test
What are the steps for a pooled two-sample t-test?
STEP 1: Write the hypotheses
H0 : μx = μy
Where μx and μy 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
H1 : μx < μy or H1 : μx > μy or H1 : μx ≠ μy
The alternative hypothesis will depend on what is being tested (see sections for one-tailed and two-tailed tests)
STEP 2: Enter the data into your GDC
Enter two lists of data – one for each sample
Choose the pooled option
Your GDC will then give you the p-value
STEP 3: Decide whether there is evidence to reject the null hypothesis
Compare the p-value with the given significance level
If p-value < significance level then reject H0
If p-value > significance level then accept H0
STEP 4: Write your conclusion
If you reject H0
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
If you accept H0
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
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One-tailed Tests
How do I perform a one-tailed t-test?
A one-tailed test is used to test one of the two following cases:
The population mean of X is bigger than the population mean of Y
The alternative hypothesis will be: H1 : μx > μy
Look out for words such as increase, bigger, higher, etc
The population mean of X is smaller than the population mean of Y
The alternative hypothesis will be: H1 : μx < μy
Look out for words such as decrease, smaller, lower, etc
If you reject the null hypothesis then
This suggests that the population mean of X is bigger than the population mean of Y
If the alternative hypothesis is H1 : μx > μy
This suggests that the population mean of X is smaller than the population mean of Y
If the alternative hypothesis is H1 : μx < μy
Worked Example
The times (in minutes) for children and adults to complete a puzzle are recorded below.
Children | 3.1 | 2.7 | 3.5 | 3.1 | 2.9 | 3.2 | 3.0 | 2.9 |
| |
Adults | 3.1 | 3.6 | 3.5 | 3.6 | 2.9 | 3.6 | 3.4 | 3.6 | 3.7 | 3.0 |
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.
a) Write down the null and alternative hypotheses.
b) Find the p-value for this test.
c) State whether the creator’s claim is supported by the test. Give a reason for your answer.
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Two-tailed Tests
How do I perform a two-tailed t-test?
A two-tailed test is used to test the following case:
The population mean of X is different to the population mean of Y
The alternative hypothesis will be: H1 : μx ≠ μy
Look out for words such as change, different, not the same, etc
If you reject the null hypothesis then
This suggests that the population mean of X is different to the population mean of Y
You can not state which one is bigger as you were not testing for that
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
Worked Example
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.
Studies French | 61 | 82 | 77 | 80 | 99 | 69 | 75 | 71 | 81 |
Studies Spanish | 74 | 79 | 83 | 66 | 95 | 79 | 82 | 81 | 85 |
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.
a) Write down the null and alternative hypotheses.
b) Find the p-value for this test.
c) Write down the conclusion to the test. Give a reason for your answer.
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