Reading and Comprehension Questions for Chapter 10
1. This is an appropriate statement of a statistical hypothesis: .
True False
False – hypotheses involve population parameters not sample statistics.
2. Completely randomized experiments are necessary to perform comparative studies and determine cause-and-effect relationships.
True False
False
3. In testing hypotheses to compare the means of two populations, it does not matter if the two populations have different variances.
True False
True
4. If a 95% CI on the difference in two means is you would have rejected the null hypothesis at the 0.05 level of significance.
True False
True – recall the relationship between hypothesis tests and CIs.
5. In the two-sample pooled t-test the pooled variance is computed by averaging the two individual sample variances.
True False
False - The pooled estimate of variance is only an average of the two sample variances if the sample sizes are equal.
6. The assumptions for the two-sample pooled t-test for the difference in two means are that there are random samples from two independent normal populations with common but unknown variance.
True False
True
7. If are the means of two random samples of sizes from independent normal populations with means and unknown but common variances, then the statistic
has a t distribution with degrees of freedom:
a.
b.
c.
d. none of the above.
Answer – c.
8. When comparing the means of two independent normal populations, if the two population variances are unequal, the pooled t-test can still be used.
True False
False – this is case 2 of the two-sample t-test and requires a different test procedure.
9. A normal probability plot of the observations from both samples can be useful in verifying the assumptions for the two-sample t-test and in determining whether the assumption of equal variances is reasonable.
True False
True –
10. The assumption of sampling from normal populations is very important for the validity of the t-test.
True False
False – In general, t-tests are quite robust to the assumption of normality.
11. The number of degrees of freedom for the two-sample t-testare the same regardless of whether or not the two population variances are equal.
True False
False – both the degrees of freedom and the test statistics are different.
12. In a two-sample pooled t-testwith 24 degrees of freedom, the computer value of the test statistic is t0 = 2.55. If the alternative hypothesis is two-sided, and if , then the P-value of this test is:
a. P > 0.01
b. P < 0.01
c. P < 0.02
d. 0.02 < P < 0.05
Answer – c. Because t0 = 2.55 > 2.49 and the test is two-sided, the P-value must be less that 2(0.01) = 0.02.
13. Consider the following display:
Power and Sample Size
2-Sample t Test
Testing mean 1 = mean 2 (versus not =)
Calculating power for mean 1 = mean 2 + difference
Alpha = 0.05 Assumed standard deviation = 2
Sample
Difference Size Power
2 15 0.752923
The sample size is for each group.
If the sample size had been smaller, the power would have been smaller.
True False
True – if the difference between means is fixed and the standard deviation is the same, the power will decrease as the sample size decreases.
14. Consider the information from question 13. If the standard deviation had been larger, the power would have been larger.
True False
False – If the standard deviation had been larger the difference between means would have been smaller in standard deviation units, thus making the difference between means harder to detect. Therefore the power of the test would have been smaller.
15. A 95% confidence interval on the different in two means is . Which of the following statements is false?
a. If the sample sizes had been larger the 95% CI would have been shorter.
b. A 99% CIU would be longer.
c. The null hypothesis in is rejected at the one percent level of significance.
d. The null hypothesis in is not rejected at the five percent level of significance.
Answer – c. The null hypothesis is rejected at the five percent level of significance.
16. The paired t-test should be considered when different experimental units are not very similar and at least two observations can be collected on each experimental unit.
True False
True
17. An F random variable is defined as the ratio of two independent chi-square random variables each divided by their numbers of degrees of freedom.
True False
True
18. The F distribution is used to conduct hypothesis tests and construct confidence intervals about the variances of two independent normal populations.
True False
True
19. The test statistic for is where the are the sample variances of two random samples from independent normal populations.
True False
True
20. A 95% CI on the ratio of the variances of two independent normal populations is . Consequently, the null hypothesis is rejected at the five percent level of significance.
True False
False – the CI includes the value unity, so would not be rejected at the five percent level.
21. Tests of hypotheses and CIs on variances are more sensitive to the normality assumption that tests and CIs on means.
True False
True
22. Statistical tests and CIs on the difference in two proportions can be performed by using the normal approximation to the binomial distribution.
True False
True
23. In testing the computed value of the test statistic was z0 = 2.0. The P-value for this test is less than 0.025.
True False
True – the normal percentile with 0.025 to the right is 1.96, and since z0 > 1.96 the P-value must be less that 0.025.
24. Consider the following information:
Power and Sample Size
Test for Two Proportions
Testing proportion 1 = proportion 2 (versus not =)
Calculating power for proportion 2 = 0.2
Alpha = 0.05
Sample
Proportion 1 Size Power
0.1 25 0.165118
The sample size is for each group.
A smaller value for proportion 2 would have led to higher power.
True False
False – A smaller value for proportion 2 would have made the difference in proportions even smaller and harder to detect with samples of size 25. This would have resultedin lower power of the test.
25. The Wilcoxon rank-sum test is a test for comparing the means of two independent continuous distributions.
True False
True
26. The normal approximation for the Wilcoxon rank-sum test requires that both sample sizes be at least
a. 20
b. 50
c. 8
d. None of the above.
Answer – c.