Hypothesis Testing for Mean (Two Sample) (DP IB Maths: AI HL)

What is a two-sample test?

• A two-sample test is used to compare the means (μ₁ & μ₂) of two normally distributed populations
• You use a z-test when the population variances ( & ) are known
• You use a t-test when the population variances are unknown
• In this course you will assume the variances are equal and use a pooled sample for a t-test
• In a pooled sample the data from both samples are used to estimate the population variance

What are the steps for performing a two-sample test on my GDC?

• STEP 1: Write the hypotheses
• H₀ : μ₁= μ₂
• Clearly state that μ₁ & μ₂ represent the population means
• Make sure you make it clear which mean corresponds to which population
• In words this means that the two population means are equal

• For a one-tailed test H₁ : μ₁ < μ₂ or H₁ : μ₁ > μ₂
• For a two-tailed test: H₁ : μ₁ ≠ μ₂
• The alternative hypothesis will depend on what is being tested
• STEP 2: Decide if it is a z-test or a t-test
• If the populations variances are known then use a z-test
• If the populations variances are unknown then use a t-test
• Assume the variances are equal and use a pooled sample
• STEP 3: Enter the data into your GDC and choose two-sample z-test or two-sample t-test
• If you have the raw data
• Enter the data as a list
• Enter the values of σ₁ & σ₂ if a z-test
• Choose the pooled option if a t-test
• If you have summary statistics (only for a z-test)
• Enter the values of , , σ₁, σ₂, n₁ & n₂
• Your GDC will give you the p-value
• STEP 4: Decide whether there is evidence to reject the null hypothesis
• If the p-value < significance level then reject H₀
• STEP 5: Write your conclusion
• If you reject H₀­ then there is evidence to suggest that...
• The mean of the 1^st population is smaller (for H₁ : μ₁ < μ₂)
• The mean of the 1^st population is bigger (for H₁ : μ₁ > μ₂)
• The means of the two populations are different (for H₁ : μ₁ ≠ μ₂)
• If you accept H­₀ then there is insufficient evidence to reject the null hypothesis which suggests that...
• The mean of the 1^st population is not smaller (for H₁ : μ₁ < μ₂)
• The mean of the 1^st population is not bigger (for H₁ : μ₁ > μ₂)
• The means of the two populations are not different (for H₁ : μ₁ ≠ μ₂)

What is a paired t-test?

• A paired test is where you take two samples but each data point from one sample can be paired with a data point from the other sample
• These are used when one group of members are used twice and the two results for each member are paired
• It could be to compare the sample before and after introducing a new factor
• It could be to compare the sample under two different conditions
• For this test you use the differences between the pairs and treat them as one sample
• As the variance of the differences is unlikely to be known you will use a t-test
• For a paired test you need to assume the differences are normally distributed
• You don’t need to assume the populations are normally distributed

What are the steps for performing a paired t-test on my GDC?

• STEP 1: Write the hypotheses
• H₀ : μ_D = 0
• Clearly state that μ_D represents the population mean of the differences
• In words this means the population mean has not changed

• For a one-tailed test H₁ : μ_D < 0 or H₁ : μ_D > 0
• For a two-tailed test: H₁ : μ_D ≠ 0
• The alternative hypothesis will depend on what is being tested
• STEP 2: Enter the data into your GDC and choose the one-sample t-test
• Enter the differences as a list
• Be consistent with the order in which you subtract paired values
• Your GDC will give you the p-value
• STEP 3: Decide whether there is evidence to reject the null hypothesis
• If the p-value < significance level then reject H₀
• STEP 4: Write your conclusion
• If you reject H₀­ then there is evidence to suggest that...
• The mean has decreased (for H₁ : μ_D < 0)
• The mean has increased (for H₁ : μ_D > 0)
• The mean has changed (for H₁ : μ_D ≠ 0)
• If you accept H­₀ then there is insufficient evidence to reject the null which suggests that...
• The mean has not decreased (for H₁ : μ_D < 0)
• The mean has not increased (for H₁ : μ_D > 0)
• The mean has not changed (for H₁ : μ_D ≠ 0)