Exam 13: Inference About Comparing Two Populations

A calculus professor wanted to test whether the grades on calculus test were the same for upper and lower classmen. The professor took a random sample of size 12 from each group. For this situation, the professor should use a matched pairs t-test.

NARRBEGIN: Anioxidants Antioxidants A food processor wants to compare two antioxidants for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with antioxidant A and 16 are treated with antioxidant B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below Antioxdeant A Antioxadant B Sample Mean 108.7 hours 98.7 hours Sample Standard Deviation 10.5 hours 13.6 hours NARREND -{Antioxidants Narrative} Develop the 95% confidence interval estimate of the ratio of the two population variances.

If some natural relationship exists between each pair of observations that provides a logical reason to compare the first observation of sample 1 with the first observation of sample 2, the second observation of sample 1 with the second observation of sample 2, and so on, the samples are referred to as:

NARRBEGIN: Aptitude Test Scores Aptitude Test Scores Two random samples of 40 students were drawn independently from two populations of students. Assume their aptitude tests are normally distributed (total points = 100). The following statistics regarding their scores in an aptitude test were obtained: $\bar { x } _ { 1 } = 76 , s _ { 1 } = 8 , \bar { x } _ { 2 } = 72 , \text { and } s _ { 2 } = 6.5$ .NARREND -{Aptitude Test Scores Narrative} Estimate with 95% confidence the difference between the two population means.

In testing the difference between two population means for which the population variances are unknown and not assumed to be equal, two independent samples are drawn from the populations. Which of the following tests is appropriate?

When the population variances are unequal, we estimate each population variance with its sample variance. Hence, the unequal-variances test statistic of $\mu _ { 1 } - \mu _ { 2 }$ is approximately Student t-distributed with n₁ + n₂ - 2 degrees of freedom.

NARRBEGIN: Speed Limits Violation Speed Limits Violation Do out-of-state motorists violate the speed limit more frequently than in-state motorists? This vital question was addressed by the highway patrol in a large eastern state. A random sample of the speeds of 2,500 randomly selected cars was categorized according to whether the car was registered in the state or in some other state and whether or not the car was violating the speed limit. The data follow. In State Cars Out af State Cars Speeding 521 328 Nat speeding 1141 510 NARREND -{Speed Limits Violation Narrative} Estimate with 95% confidence the difference in population proportions.

When the sample sizes are equal, the pooled variance of the two samples is the ____________________ of the two sample variances.

NARRBEGIN: Engine Wear Engine Wear To compare the wearing of two types of automobile engines, 1 and 2, an experimenter chose to "pair" the measurements, comparing the wear for the two types of engines on each of 7 automobiles, as shown below. Autamobile 1 2 3 4 5 6 7 Engine 1 15 7 9 10 13 11 Engine 2 12 18 8 9 12 11 10 NARREND -{Engine Wear Narrative} Estimate with 90% confidence the mean difference and interpret.

To estimate the ratio of the population variances you use the ____________________ of the ____________________ variances.

When testing for the equality of two population variances the number in the null hypothesis is ____________________.

Which of the following statements is correct regarding the percentile points of the F-distribution?

NARRBEGIN: Clothing Expenditures Clothing Expenditures A marketing consultant was in the process of studying the perceptions of married couples concerning their monthly clothing expenditures. He believed that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of ten married couples and asks each spouse to estimate the family clothing expenditure (in dollars) during the previous month. The data are shown below. Couple Husband Wife 1 380 270 2 280 300 3 215 185 4 350 320 5 210 180 6 410 390 7 250 250 8 360 320 9 180 170 10 400 330 NARREND -{Clothing Expenditures Narrative} Estimate with 95% confidence the population mean difference.

The expected value of $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is $E \left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right) = \mu _ { 1 } - \mu _ { 2 }$ .

NARRBEGIN: TV Sex TV Sex A survey of 1,500 Canadians reveals that 945 believe that there is too much sex on television. In a survey of 1,500 Americans, 810 believe that there is too much television sex.NARREND -A councilwoman regularly polls her constituency to gauge her level of support among voters. This month, 652 out of 1158 voters support her. Five months ago, 412 out of 982 voters supported her. With a 5% significance level, can she infer that support has increased by at least 10 percentage points?

In testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is normal if the sample sizes are both greater than 30.

NARRBEGIN: Anioxidants Antioxidants A food processor wants to compare two antioxidants for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with antioxidant A and 16 are treated with antioxidant B, and the number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below Antioxdeant A Antioxadant B Sample Mean 108.7 hours 98.7 hours Sample Standard Deviation 10.5 hours 13.6 hours NARREND -{Antioxidants Narrative} State the null and alternative hypotheses for determining if the population variances differ for Antioxidants A and B.

The sampling distribution of $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n₁ and n₂ are large.

Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t-test statistic is 44.

NARRBEGIN: Undergraduates' Test Scor Undergraduates' Test Scores 35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random sample of 50 students who only completed high school, the mean and standard deviation of the test scores were 72.1 and 14.6, respectively.NARREND -{Undergraduates' Test Scores Narrative} Estimate with 90% confidence the difference in mean scores between the two groups of students.