Chapter 10 Hypothesis Testing On The Comparisons of Two Population Means

Chapter Outline Inferences About the Difference Between Two Population Means: s 1 and s 2 Known

Interval Estimation of m 1 – m 2 Inferences About the Difference Between Two Population Means: s 1 and s 2 Known Interval Estimation of m 1 – m 2 Hypothesis Tests About m 1 – m 2

Estimating the Difference Between Two Population Means So far, we have studied the estimation of ONE population mean based on the distribution of a sample drawn from the population. Similarly, when we want to estimate the difference between two population means, we can pick a sample from each population and analyze the sampling distribution of the difference between the two sample means. More specifically, to estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2. The point estimator of the difference between the means of the populations 1 and 2 is simply .

Sampling Distribution of Expected Value Standard Deviation (Standard Error) where: 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2

Interval Estimation of 1 - 2: s 1 and s 2 Known Interval Estimate where: 1 -  is the confidence coefficient

Interval Estimation of 1 - 2: s 1 and s 2 Known Example: Textbook Spending To compare the amount of money students spent on textbooks in the Fall semester between two universities, independent random samples were chosen and revealed the following sample information: University A University B Sample Size 50 40 Sample Mean $260 $250 We assume that the population standard deviations of students’ spending on textbooks are known, i.e. s A=$20 and s B=$23 .

Interval Estimation of 1 - 2: s 1 and s 2 Known Example: Textbook Spending Let us develop a 95% confidence interval estimate of the difference in the average spending on textbooks between the two universities. Point estimate of 1 - 2 = = 260 - 250 = $10 where: 1 = average spending on textbooks of students in University A 2 = average spending on textbooks of students in University B

Interval Estimation of 1 - 2: s 1 and s 2 Known Example: Textbook Spending 10 + 9.03 or $0.97 to $19.03 We are 95% confident that the difference in the average spending on textbooks by students between the two universities is from $0.97 to $19.03.

Hypothesis Tests About 1 - 2: s 1 and s 2 Known Hypotheses Left-tailed Right-tailed Two-tailed Test Statistic

Hypothesis Tests About 1 - 2: s 1 and s 2 Known Example: Textbook Spending Can we conclude, using  = .05, that, on the average, students at University A spent the same amount on textbooks as the students at University B?

Hypothesis Tests About 1 - 2: s 1 and s 2 Known p-Value and Critical Value Approaches 1. Develop the hypotheses. H0: 1 - 2 = 0  Ha: 1 - 2  0 where: 1 = average spending on textbooks by students in university A 2 = average spending on textbooks by students in university B 2. Specify the level of significance. a = .05

Hypothesis Tests About 1 - 2: s 1 and s 2 Known p-Value and Critical Value Approaches 3. Compute the value of the test statistic:

Hypothesis Tests About 1 - 2: s 1 and s 2 Known p-Value Approach 4. Compute the p-value: For z=2.17, the area to the left of 2.17 is 1-0.985=0.015. So, the p-value is 2(0.015)=0.03 for the two-tailed test. Determine whether to reject H0: Because p-value < a = .05, we reject H0. At the .05 level of significance, the sample data indicate that the average spending on textbooks by students in university A is NOT the same at that in university B.

Hypothesis Tests About 1 - 2: s 1 and s 2 Known Critical Value Approach 4. Determine the critical value and rejection rule: For  = .05, z0.05/2= z0.025 = 1.96. Reject H0 if z > 1.96 or z < -1.96 5. Determine whether to reject H0: Because the test statistic z = 2.17 > 1.96, we reject H0. Again, at the .05 level of significance, the sample data indicate that the average spending on textbooks by students in university A is NOT the same at that in university B.