Exam 10: Two-Sample Hypothesis Tests

The Welch-Satterthwaite test is more conservative than the pooled variance test to compare two population means with unknown variances in independent samples.

The table below shows the mean number of daily errors by seven air traffic controller trainees during the first two weeks on the job. We want to perform a paired t-test at ? = .05 to see if the mean daily errors have decreased from Week 1 to Week 2. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Trainee }$ T1 T2 T3 T4 T5 T6 T7 Week 1 5.1 3 12.1 6.2 11.5 7.8 2.2 Week 2 3.2 2.2 8.7 7.7 9.4 7.8 3.1 The right-tailed critical value at ? = 0.05 is:

In a test of a new surgical procedure, the five most respected surgeons in FlatBroke Township were invited to Carver Hospital. Each surgeon was assigned two patients of the same age, gender, and overall health. One patient was operated upon in the old way, and the other in the new way. Both procedures are considered equally safe. The surgery times are shown below: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Surgeon }$ Allen Bob Chloe Daphne Edgar Old way 36 55 28 40 62 New way 31 45 28 35 57 Which test should we use to test for zero difference in mean times?

A random sample of Ersatz University students revealed that 16 females had a mean of $22.30 in their wallets with a standard deviation of $3.20, while 6 males had a mean of $17.30 with a standard deviation of $9.60. The value of the test statistic for a folded F-test for equal variances is:

Group 1 has a mean of 13.4 and group 2 has a mean of 15.2. Both populations are known to have a variance of 9.0 and each sample consists of 18 items. What is the test statistic to test for equality of population means?

The table below shows the mean number of daily errors by air traffic controller trainees during the first two weeks on the job. We want to perform a paired t-test at ? = .05 to see if the mean daily errors decreased significantly. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Trainee }$ T1 T2 T3 T4 T5 T6 T7 Week 1 5.1 3 12.1 6.2 11.5 7.8 2.2 Week 2 3.2 2.2 8.7 7.7 9.4 7.8 3.1 The test statistic is:

An F-test for equality of variances gives a p-value of .003. At α = .05, what conclusion can be made about the preferred test to compare the means for the same sample?

If the sample proportions are p₁ = 6/90 and p₂ = 4/100, normality may be assumed in a test comparing the two population proportions.

Which is not a type of comparison for which you would anticipate a two-sample test?

A random sample of Ersatz University students revealed that 16 females had a mean of $22.30 in their wallets with a standard deviation of $3.20, while 16 males had a mean of $17.30 with a standard deviation of $9.60. In comparing the population variances at α = .10 in a two-tailed test, we conclude that:

If the sample proportions are p₁ = 15/60 and p₂ = 20/90, normality may be assumed in a test comparing the two population proportions.

The test statistic in an F test for equal variances is the ratio of the sample variances.

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 4.389 1.874 Sample size 7 10 At ? = .05, can you conclude that the first sample has a larger variance than the second sample?

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. To test for equality in the proportions of men versus women who experienced improvement, the test statistic is approximately:

Nacirema Airlines is buying a fleet of new fuel-efficient planes. The HogJet and the LitheJet both meet their price and performance needs, and both planes meet EPA noise guidelines. However, the quieter plane is preferred. Each plane is flown through a typical takeoff and landing sequence 10 times, while remote sensors at ground level record the noise levels (in decibels). The table below summarizes the sound level tests using Excel's default level of significance (? = 0.05). $\quad$$\quad$$\text { t-Test: Assuming Equal Variances }$ $\quad$$\quad$$\quad$$\quad$ $\text { t-Test: Assuming Unequal Variances }$ takeJet HogJet Mean 80.3368 82.4669 Variance 0.7178 4.7385 Observations 10 10 Pooled Variance 2.7282 Hypothesized Diff 0.0000 df 18 t Stat -2.8837 P(T eat) one-tail 0.0049 t Critical one-tail 1.7341 P(T =-t) two-tail 0.0099 t Critical two-tail 2.1009 takeJet HogJet Mean 80.3368 82.4669 Variance 0.7178 4.7386 Observations 10 10 Hypothesized Diff 0 df 12 t Stat -2.8837 P(T ett ) one-tail 0.0069 t Critical one-tail 1.7823 (<=) two-tail 0.0137 Critical two-tail 2.1788 In a left-tailed test comparing the means at ? = .05, we would:

Mary analyzed occupancy rates at two community hospitals and obtained the following Excel results. t-Test for Acute Care Occupancy Rates MaxHealth HealthPro Mean 62.5462 68.4800 Variance 108.2377 98.3707 Observations 13 10 Hypothesized Diff 0 df 20 t Stat -1.3923 P(T<=t) one-tail 0.0896 t Critical one-tail 1.7247 P(T<=t) two-tail 0.1791 t Critical two-tail 2.0860 Which conclusion is correct in a two-tailed test at ? = .05?

A psychology researcher has a theory that predicts women will tend to carry more cash than men. A random sample of Ersatz University students revealed that 16 females had a mean of $22.30 in their wallets with a standard deviation of $3.20, while 16 males had a mean of $17.30 with a standard deviation of $9.60. The researcher's hypothesis would lead us to perform a:

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. In a right-tailed test, the p-value is .0275, so at α = .025 we should:

Nacirema Airlines is buying a fleet of new fuel-efficient planes. The HogJet and the LitheJet both meet their price and performance needs, and both planes meet EPA noise guidelines. However, the quieter plane is preferred. Each plane is flown through a typical takeoff and landing sequence 10 times, while remote sensors at ground level record the noise levels (in decibels). The table below summarizes the sound level tests using Excel's default level of significance (? = 0.05). $\quad$$\quad$$\text { t-Test: Assuming Equal Variances }$ $\quad$$\quad$$\quad$$\quad$ $\text { t-Test: Assuming Unequal Variances }$ takeJet HogJet Mean 80.3368 82.4669 Variance 0.7178 4.7385 Observations 10 10 Pooled Variance 2.7282 Hypothesized Diff 0.0000 df 18 t Stat -2.8837 P(T eat) one-tail 0.0049 t Critical one-tail 1.7341 P(T =-t) two-tail 0.0099 t Critical two-tail 2.1009 takeJet HogJet Mean 80.3368 82.4669 Variance 0.7178 4.7386 Observations 10 10 Hypothesized Diff 0 df 12 t Stat -2.8837 P(T ett ) one-tail 0.0069 t Critical one-tail 1.7823 (<=) two-tail 0.0137 Critical two-tail 2.1788 After inspecting this table, we would most likely:

In a test for equality of two proportions, the sample proportions were p₁ = 12/50 and p₂ = 18/50. The pooled proportion is: