Exam 10: Two-Sample Hypothesis Tests

During a test period, an experimental group of 10 vehicles using an 85 percent ethanol-gasoline mixture showed mean CO₂ emissions of 667 pounds per 1000 miles, with a standard deviation of 20 pounds. A control group of 14 vehicles using regular gasoline showed mean CO₂ emissions of 679 pounds per 1000 miles with a standard deviation of 15 pounds. At α = 0.05, in a left-tailed test (assuming equal variances) the test statistic is:

John wants to compare two means. His sample statistics were $\bar { x } _ { 1 } = 22.7 , s _ { 1 } ^ { 2 } = 5.4 , n _ { 1 } = 9$ and $\bar { x } _ { 2 } = 20.5 , s _ { 2 } ^ { 2 } = 3.6 , n _ { 2 } = 9$ . Assuming equal variances, the 95 percent confidence interval for the difference of the population means is approximately:

A pooled proportion is calculated by giving each sample proportion an equal weight.

A certain psychological theory predicts that men want bigger families than women. Kate asked each student in her psychology class how many children he or she considered ideal for a married couple and obtained the Excel results shown below at ? = .05. $\text { t-Test: Two-Sample Assuming Equal Variances }$ $\text { t-Test: Two-Sample Assuming Unequal Variances }$ Men Mean 2.4571 Variance 0.7261 Observations 35 Pooled Variance 0.59206 Hypothesized Diff 0 df 48 t Stat 1.364 P(Tet) one-tail 0.090 t Critical one-tail 1.677 P(T-t) two-tail 0.179 t Critical fwo-tail 2.011 Women. 2.1333 Mean 0.2667 Variance 15 Observations Hypothesized Diff df 1 Stat P(Tet) one-tail 1 Critical one-tail P(T <= t) two-tail 1 Critical two-tail Men Women. 2.4571 2.1333 0.7261 0.2667 35 15 0 42 1.650 0.053 1.682 0.106 2.018 What conclusion can you draw in a two-tailed test at ? = .05?

Two well-known aviation training schools are being compared using random samples of their graduates. It is found that 70 of 140 graduates of Fly-More Academy passed their FAA exams on the first try, compared with 104 of 260 graduates of Blue Yonder Institute. To compare the two proportions, the assumption of normality of the test statistic is:

When sample data occur in pairs, an advantage of choosing a paired t-test is that it tends to increase the power of a test, as compared to treating each sample independently.

Does the Speedo Fastskin II Male Hi-Neck Bodyskin competition racing swimsuit improve a swimmer's 200-yard individual medley performance times? A test of 100 randomly chosen male varsity swimmers at several different universities showed that 66 enjoyed improved times, compared with only 54 of 100 female varsity swimmers. In comparing the proportions of males versus females, is it safe to assume normality?

Two well-known aviation training schools are being compared using random samples of their graduates. It is found that 70 of 140 graduates of Fly-More Academy passed their FAA exams on the first try, compared with 104 of 260 graduates of Blue Yonder Institute. To compare the pass rates, find the critical value for a right-tailed test at α = .05.

Of 200 youthful gamers (under 18) who tried the new Z-Box-Plus game, 160 rated it "excellent," compared with only 144 of 200 adult gamers (18 or over). The 95 percent confidence interval for the difference of proportions would be approximately:

In a left-tailed test comparing two means with unknown variances assumed to be equal, the test statistic was t = -1.81 with sample sizes of n₁ = 8 and n₂ = 12. The p-value would be:

In a random sample of patient records in Cutter Memorial Hospital, six-month postoperative exams were given in 90 out of 200 prostatectomy patients, while in Paymor Hospital such exams were given in 110 out of 200 cases. In a left-tailed test for equality of proportions, the test statistic is:

Carver Memorial Hospital's surgeons have a new procedure that they think will decrease the variance in the time it takes to perform an appendectomy. A sample of 8 appendectomies using the old method had a variance of 36 minutes, while a sample of 10 appendectomies using the experimental method had a variance of 16 minutes. At α = .10 in a two-tailed test for equal variances, the critical values are:

We could use the same data set for two independent samples (i.e., two columns of data) either to compare the means (t-test) or to compare the variances (F test).

Litter sizes (number of pups) for randomly chosen dogs from two breeds were compared. The sample data were entered into Excel, and the following results were produced. t-Test: Two-Sample Assuming Unequal Variances Dalmation Labrador Mean 4.8125 5.461538 Variance 3.4958333 0.602564 Observations 16 13 Hypothesized Mean Diff 0 df 21 t Stat -1.261183 What is the critical value for a left-tailed test comparing the means at ? = .05?

At Huge University, a sample of 200 business school seniors showed that 26 planned to pursue an MBA degree, compared with 120 of 800 arts and sciences seniors. We want to know if the proportion is higher in the arts and sciences group. What is the z test statistic?

Two well-known aviation training schools are being compared using random samples of their graduates. It is found that 70 of 140 graduates of Fly-More Academy passed their FAA exams on the first try, compared with 104 of 260 graduates of Blue Yonder Institute. To compare the pass rates, the pooled proportion would be:

A paired t-test with two columns of 10 observations in each column would use d.f. = 18.

When using independent samples to test the difference between two population means, a pooled variance is used if the population variances are unknown and assumed equal.

In a right-tailed test comparing two means with known variances, the sample sizes were n₁ = 8 and n₂ = 12. At α = .05, the critical value would be:

The table below shows two samples taken to compare the mean age of individuals who purchased the iPhone 3G at two AT&T store locations. Statistic Ann Arbor Livonia Mean 25.817 31.248 St Dev 3.389 1.874 Sample size 7 10 What are the critical values for a two-tailed test for equal variances at ? = .05?