Hypothesis Testing for Mean (Two Sample) (DP IB Applications & Interpretation (AI)): Revision Note

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

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)

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

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)