IB DP Maths: AI HL复习笔记4.12.2 Hypothesis Testing for Mean (Two Sample)

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 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)
  • 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_Drepresents the population mean of the differences
    • In words this means the population mean has not changed

• • For a one-tailed test H₁ : μ_D<0or 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 H­₀then 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)