Comparing Two Populations Section 11.2.1 Lesson11.2.1

Starter 11.2.1 Write the assumptions that underlie the use of the t procedures. Which assumption is so important you can’t work without it? What is the best way to tell if the distribution assumption is met?

Starter Solution The two main assumptions we need are: The sample came from a valid SRS of the population. The population is approximately normally distributed. We can’t do anything without a valid SRS. Plot the data to see if they are approximately normal. Note that if sample size is large we can live with skewness or outliers.

Today’s Objectives Students will compare the means of two populations to determine whether the differences are statistically significant. Students will perform a two-sample t-test from formula. Students will form a two-population confidence interval from formula. California Standard 20.0 Students are familiar with the t distributions and t test and understand their uses.

Two-sample Problems Used when we want to compare two different populations or experimental treatments This is how we will compare the placebo group with the treatment group We need a separate sample from each population That’s why it’s called “two-sample” The goal is to compare the population means to determine if they are different (in a statistically significant way) This is not matched pairs: we don’t have information about individual scores

Assumptions for comparing two means There are two SRS’s from distinct populations Both populations are normally distributed. We do not know μ and σ. With large sample sizes, the Central Limit Theorem assures approximately normal sample means. The samples are independent One sample has no influence on the other. Matching violates independence.

Procedure Assumptions Assumptions So Far… One population t for means Two population t for means Assumptions SRS, Normal Dist SRS, Normal, Independent

Population symbols we will use We want to compare μ1 and μ2 Are they equal or different? To do compare, we use sample means. Population Variable Mean Std Dev 1 x1 μ1 σ1 2 x2 μ2 σ2

Sample symbols we will use We will use sample means Ë1 and Ë2 to draw inferences about population means μ1 and μ2 Population Sample Size Mean Std Dev 1 n1 Ë1 s1 2 n2 Ë2 s2

The distribution of Ë1 – Ë2 Shape: The distribution of all possible differences (Ë1 - Ë2) is approximately normal Center: The mean of the distribution of all possible differences (Ë1 – Ë2) is (μ1 – μ2 ) Spread: The standard error of the distribution is given by the formula

Two-sample Hypothesis Tests Establish hypotheses. Ho: μ1 = μ2 Ha: μ1 ≠ μ2 OR µ1 < μ2 OR µ1 > μ2 Calculate the t-test statistic. Formula on next slide Find the p-value from table or tcdf command. Write a three-phrase conclusion statement.

The two-sample t statistic Note: The degrees of freedom will be n-1 for the smallest sample size.

Example 11.9 A test of social abilities is given to 133 men and 162 women. Here are the results Is there a significant difference between the two populations (men and women)? Group Sex n Ë s 1 Male 133 25.34 5.05 2 Female 162 24.94 5.44

State the hypotheses Calculate the t statistic Ho: μ1 = μ2 Ha: μ1 ≠ μ2 Now apply the formula for t. You should have gotten t = 0.654 So what is P? tcdf(.654, 999, 132) = 0.257 (Remember d.f. = smallest n-1) But this is a two-tailed test, so: P = 2 x 0.257 = 0.514 Conclusion: Because p is greater than any reasonable alpha, there is not enough evidence to support the claim that men and women’s scores differ.

Two-sample Confidence Interval As before, confidence intervals take the form Estimate ± critical value x standard error So the confidence interval for μ1 – μ2 is

Example 11.9 again Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test

Example 11.9 again Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test

Example 11.9 again Calculate a 95% Confidence Interval for the mean difference between men and women on the social skills test. Notice that the interval contains zero, so this is consistent with the failure to reject in the hypothesis test.

Today’s Objectives Students will compare the means of two populations to determine whether the differences are statistically significant. Students will perform a two-sample t-test from formula. Students will form a two-population confidence interval from formula. California Standard 20.0 Students are familiar with the t distributions and t test and understand their uses.

Homework Read pages 617 – 627 Do problems 33 - 35