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AP Statistics Chapter 13: Comparing Two Population Parameters

13.1 – Comparing Two Means

Two-Sample Problems

• The goal of inference is to compare the responses to two treatments or to compare the characteristics of two populations.
• We have a separate sample from each treatment or each population.

Conditions for Comparing Two Means

• We have two SRS’s, from two distinct populations. The samples are independent. That is, one sample has no influence on the other. Matching violates independence, for example. We measure the same variable for both samples.
• Both populations are normally distributed. The means and standard deviations of the populations are unknown.
• If n1+n2 is at least 30, we do not need to worry about normality

The Two-Sample t Procedures

The form of the confidence interval for a population mean mu₁- mu₂ is

To test the hypothesis Ho: mu₁ = mu₂ based, compute the two-sample t statistic

Against either an alternate hypothesis of mu₁ < mu_2, : mu₁ > mu₂: or mu₁ ≠ mu_2.

Using the Two-Sample t Procedures

Apply the same rules concerning sample size from the one-sample t, but add the two sample sizes (n₁+n₂) before applying the rules.

13.2 – Comparing Two Proportions

Conditions for Inference for Two Proportions

• The data are a simple random sample (SRS) from the both populations of interest.
• Counts of successes and failures must be 5 or more for both proportions.
• Both population sizes are at least 10 times greater than their sample sizes
  

The Two-Proportion z Procedures

The form of the conf. interval for the difference between two population proportions p₁-p₂ is

To test the hypothesis Ho: p₁ = p₂ based on an SRS’s of size n₁ and n₂, compute the two-proportion z statistic

Note: Pc-hat in the denominator of the formula above is called the combined sample proportion. It is calculated as

Example: If P1-hat = 13/42 and P2-hat = 19/52, then Pc-hat = (13+19)/(42+52) = 32/94.