Practice exam 2 - stat PDF

Preview

Summary

stat...

Description

Practice Test for Exam2 of Stat 302, Fall 2020

1

1. Which of the following statements about P-value is correct? (a) (b) (c) (d)

As significance level α increases, so does the P-value. The P-value measures the probability that the null hypothesis is true. The P-value measures the probability of making a Type I error. An extremely small P-value indicates that the actual data is very different from that expected if the null hypothesis is true. (e) The larger the P-value, the stronger the evidence against the null hypothesis.

2. A test of H0 : µ = 0 vs. Ha : µ > 0 is conducted on the same population independently by two different researchers. Each researcher takes a simple random sample from the population. They both use the same sample size and the same level of significance. Which of the following statements must be true about their results of test? (a) (b) (c) (d) (e)

The power of their tests are the same given the true mean µ = 2. The values of their test statistics are the same. Their decisions about whether or not to reject the null hypothesis are the same. The P-values of their tests are the same. Their observed sample statistics are the same.

3. Suppose the P-value for testing hypotheses was computed to be 0.057. Which of the following statements is correct? (a) (b) (c) (d) (e)

We can reject the null hypothesis at α = 0.10 but not at 0.05 or 0.01. We can reject the alternative hypothesis at α = 0.01 and 0.05 but not at 0.10. We can reject the alternative hypothesis at α = 0.05 and the null at α = 0.10. We can reject the null hypothesis at α = 0.01 and 0.05 but not at 0.10. We can reject the alternative hypothesis at α = 0.10 but not at 0.05 or 0.01.

4. When testing H0 : µ = µ0 vs. Ha : µ > µ0 at α = 0.05, which of the flowing statements is (are) possible reasons for committing a Type II error? I The sample mean is near the hypothesized mean in H0. II The sample size is too small. III The standard deviation is too large. (a) (b) (c) (d) (e)

I and II I and III II and III III only All I, II, and III are true

2

5. Suppose a one-sample t-test from a sample of n = 10 observations for the hypotheses H0 : µ = 64 vs. Ha : µ 6= 64 has the test statistic t = 3.21. The P-value for this test is (a) (b) (c) (d) (e)

greater than 0.02 between 0.0025 and 0.005 between 0.005 and 0.01 between 0.01 and 0.02 not possible to be found for a negative test statistic.

6. After the first midterm of STAT302, a student believes the mean score of all students is below 80. He plans to collect a sample of 30 students to test his hypotheses H0 : µ = 80 vs. Ha : µ < 80. He determined that he will reject the null hypothesis if he obtains a sample mean that is 78.5 or lower. Suppose the true mean of STAT 302 students’ midterm score is 78. Below are the plotted sampling distributions. The bold distribution indicates the true sampling distribution and the other distribution indicates the sampling distribution if H0 were true. In which plot, the shaded area indicates the power of test?

(a) (b) (c) (d) (e)

Plot 1 Plot 2 Plot 3 Plot 4 None of the above.

3

7. The owner of travel agency would like to determine whether or not the mean age of the agency’s customers is over 24. If so, he plans to alter the destination of their special cruises and tours. If he concludes the mean age is over 24 when it is actually not, he makes a ( ) error. If he concludes the mean age is not over 24 when it actually is, he makes a ( ) error. (a) (b) (c) (d) (e)

Type II; Type II Type I; Type I Type I; Type II Type II; Type I α; Power

8. Ten healthy adult subjects were asked to walk on a treadmill with two different setting of disturbances. Each subject experienced both kinds of disturbance in random order and the number of steps until adaptation was recorded. Here are the summary data for these 10 subjects: n

mean

std.dev

Disturbance A

10

7.32

2.10

Disturbance B

10

5.17

1.86

To estimate the true difference of the two mean steps until adaptation between the two disturbance settings, the standard error used in computing the confidence interval is p p (a) 2.1/ 10p + 1.86/ 10 (b) 2.981 p ⇥ 1/10 + 1/10 (c) 2.69/ p 10 (d) 2.69 ⇥ 1/10 + 1/10 p 2 (e) (2.18) /10 + (1.86)2 /10

9. The Nielson Company reported that nationally 30% of Millennials order groceries online. Suppose that a U.S. grocery company wishes to test whether this figure is different in their local market. The test will be conducted at the 1% significance level. What is the probability that the grocery company will commit a Type I error? (a) (b) (c) (d) (e)

0.01 0.02 0.05 0.10 Not enough information

4

10. You are planning to use a sample proportion pˆ to estimate a population proportion p. Suppose a sample size of n = 100 and a conf idence level of 95% yielded a margin of error of 0.025. Which of the following will result in a larger margin of error? I Increasing the sample size while keeping the same confidence level II Decreasing the sample size while keeping he same confidence level III Increasing the confidence level while keeping the same sample size IV Decreasing the confidence level while keeping the same sample size (a) (b) (c) (d) (e)

I and IV II and IV II and III I and III None of the above

11. Suppose we are testing the hypotheses H0 : µ = 70 vs. Ha : µ > 70. Of the following sample means, which one will have the largest P-value? (Hint: draw a sampling distribution of x¯) (a) (b) (c) (d) (e)

x¯ = 70 x¯ = 72 x¯ = 72 x¯ = 68 x¯ = 69

12. Suppose researchers randomly select a sample to test the hypotheses: H0 : µ = 65 vs. Ha : µ 6= 65. If the researchers had collected 100 random samples each of size 30 and tested the hypothesis with each sample, which of the following is the best interpretation of the significance level 0.05? (a) If µ = 65 is really true, then for approximately 5 of the samples they will fail to reject the null hypothesis and for approximately 95 of the samples they will reject the null hypothesis. (b) If µ = 65 is really true, then for approximately 95 of the samples they will fail to reject the null hypothesis and for approximately 5 of the samples they will reject the null hypothesis. (c) If µ 6= 65 is really true, for approximately 5 of the samples they will fail to reject the null hypothesis and for approximately 95 of the samples they will reject the null hypothesis. (d) If µ 6= 65 is really true, then for approximately 95 of the samples they will fail to reject the null hypothesis and for approximately 5 of the samples they will reject the null hypothesis. (e) We cannot say unless we are given the true value of µ when Ha is true.

5

13. A random sample of 25 college males was obtained and each was asked to report their actual height and what they wished as their ideal height. A 95% confidence interval for µd = mean difference between ideal and actual heights of male students in this college was 0.8 to 2.2 inches. Based on this interval, at the significance level α = 0.05, which one of the null hypotheses below (versus a two-sided alternative) can be rejected? (a) (b) (c) (d) (e)

H0 : µd = 0.5 H0 : µd = 1.0 H0 : µd = 1.5 H0 : µd = 1.8 H0 : µd = 2.0

14. Which statement regarding the power of a significance test is correct? (a) (b) (c) (d) (e)

As the standard deviation increases, the power of the test increases. As the sample size increases, the power of the test increases. As the probability of Type II error β increases, the power of the test increases. As the significance level α increases, the power of the test decreases. As the effect size increases, the power of the test decreases

15. Is there a difference in the proportion of individuals with food allergies between young children (pC ) and adults (pA )? We plan to test the following hypotheses: H0 : pC = pA vs. Ha : pC 6= pA . A survey f inds that 13 among a random sample of 184 young children have some food allergy, compared with 6 out of 163 adults. If the null hypothesis were true, the point estimate for the same but unknown population proportion in the two groups would be equal to (a) (b) (c) (d) (e)

(1/2) ⇥ (19/347) 13/184 + 6/163 19/347 13/184  6/163 (1/2) ⇥ (13/184 + 6/163)

A recent Pew research study reports, “In 2015, 17% of all U.S. newlyweds had a spouse of a different race or ethnicity, marking more than a fivefold increase since 1967, according to a new Pew Research Center analysis.” Later, they report that the margin of error for the data from 2015 was 4 percentage points (m = 0.04 for the confidence interval) with 95% confidence.

16. Interpret the margin of error for the study above. (a) The estimated maximum distance (with 95% confidence) between the sample and population proportions is 0.04. (b) The typical difference between the sample and population proportions is 0.04. 6

(c) The confidence interval will fail to capture the population proportion 4% of the time, in repeated sampling. (d) There is a 4% chance that the population proportion is not in this interval. (e) The confidence interval will fail to capture the sample proportion 4% of the time, in repeated sampling.

17. Construct a 95% confidence interval for the proportion of all U.S. newlyweds marrying a spouse of a different race or ethnicity in 2015. (a) (b) (c) (d) (e)

(0.09, 0.25) (0.07, 0.27) (0.13, 0.21) (0.11, 0.23) (0.10, 0.24)

A survey asked residents in state A and state B whether marijuana should be made legal. Assume all the conditions are met. A 95% confidence interval for pA  pB is given by (0.11, 0.18) where pA and pB are the proportion of all residents who favor legal in state A and state B, respectively.

18. Find the point estimate that was used in computing the given confidence interval (0.11, 0.18). (a) (b) (c) (d) (e)

0.29 0.145 0.07 0.035 0

19. Based on this 95% confidence interval, what can we conclude about the proportion of residents who favor legalization in state A versus state B? (a) Since all of the values in the confidence interval are greater than 0, we can conclude that the proportion in favor of legalization was greater in state A than it was in state B. (b) Since all of the values in the confidence interval are less than 1, we can conclude that there is a significant difference between the proportion in favor of legalization in state B and the proportion in favor of legalization in state A. (c) Since all of the values in the confidence interval are less than 1, we are unable to conclude that there is a significant difference between the proportion in favor of legalization in state B and the proportion in favor of legalization in state A. (d) Since all of the values in the confidence interval are greater than 0, we can conclude that the proportion in favor of legalization was greater in state B than it was in state A. (e) None of the above

7

A study randomly assigned patients who had suffered stroke to an aspirin treatment or a placebo treatment. The number of deaths due to heart attack was recorded during a 3 years follow-up period. The results of statistical analyses are summarized below. Sample

x

n

Sample proportion

placebo 27 688

0.0392

aspirin

0.0144

10 696

Difference=Placebo-Aspirn Estimate for difference: 0.0248 95% CI for difference: (0.0079, 0.0419) Test for Difference=0 (vs. 6= 0): z = 2.86, p-value=0.0041 20. Compute the margin of error corresponding to the above 95% confidence interval. (a) (b) (c) (d) (e)

4.98% 3.40% 2.49% 1.70% 0.95%

21. Explain how to interpret the p-value for the test. (a) For patients who had suffered stroke, their chance of heart attack is only 0.0041 if they would take aspirin. (b) If aspirin has no effect to heart attack for all patients who had suffered stroke, the chance to see a difference at least this extreme between the two sample proportions is only 0.0041. (c) If aspirin has no effect to heart attack for all patients who had suffered stroke, the chance to see a exactly same difference between the two sample proportions is only 0.0041. (d) If aspirin has an effect to heart attack for all patients who had suffered stroke, the chance to see a difference at least this extreme between the two sample proportions is only 0.0041. (e) If aspirin has an effect to heart attack for all patients who had suffered stroke, the chance to see a exactly same difference between the two sample proportions is only 0.0041.

8

Researchers were interested in investigating the link between energy spent on daily activity and obesity. Forty healthy participants who don’t exercise were chosen randomly from two groups: 20 from a group of normal weight individuals and 20 from a group of mildly obese but still healthy individuals. The total time in minutes over a week the participants spent standing or walking were recorded by a fitness sensors they carry. The researchers were interested in testing whether normal weight individuals stand and walk longer time than obese individuals, on average.

22. Assume all the assumptions and conditions are satisfied. Which of the following is the most appropriated statistical procedure to analyze the data? (a) (b) (c) (d) (e)

two independent samples t-test two independent proportions z-test one sample t-test any t-test will work two dependent samples paired t-test

23. The P-value for the researchers’ test was approximately zero. What conclusion should be made in context? (a) The researchers don’t have evidence that the normal and obese groups stands and walks the same amount of time, on average. (b) The researchers don’t have evidence that the normal group stands and walks longer time than the obese group, on average. (c) The researchers have evidence that the normal group stands and walks longer time than the obese group, on average. (d) The researchers have evidence that the normal and obese groups stands and walks the same amount of time, on average. (e) The P-value is too small to make a conclusion.

9

A physical therapist wanted to know whether the mean step pulse of men and women were different. Each subject was required to step up and down onto a 6-inch platform for 3 minutes. The pulse of each subject (in beats per minute) was then recorded. A 95% confidence interval for the difference in mean pulse rates (women minus men) was (1.5, 10.7) bpm.

24. Which of the following is a correct interpretation of this confidence interval? (a) We are 95% confident that pulse rates for women are between 1.5 and 10.7 beats per minute lower than for men, on average, after 3 minutes of step-ups. (b) After 3 minutes of step-ups, 95% of women’s pulse rates are between 1.5 and 10.7 beats per minute higher than men’s pulse rates. (c) We are 95 confident that pulse rates for women are between 1.5 and 10.7 beats per minute higher than for men, on average, after 3 minutes of step-ups. (d) After 3 minutes of step-ups, we are 95% confident that men’s pulse rates will be between 1.5 and 10.7 beats per minutes higher than women’s pulse rates 95% of the time. (e) After 3 minutes of step-ups, 95% of men’s pulse rates are between 1.5 and 10.7 beats per minute higher than women’s pulse rates.

25. Use this 95% confidence interval of (1.5, 10.7) to test a two-sided hypotheses if there is a difference in mean pulse rates between women and men after the exercise. Which of the following is a correct statement? (a) There is sufficient evidence at the 5% significance level to indicate that the mean pulse rate is the same for women and men after doing three minutes of step-ups because 0 is not in this confidence interval. (b) There is not sufficient evidence at the 5% significance level to indicate that the mean pulse rate is different for women and men after doing three minutes of step-ups because 0 is not in this conf idence interval. (c) There is not sufficient evidence at the 5% significance level to indicate that the mean pulse rate is the same for women and men after doing three minutes of step-ups because 0 is in this conf idence interval. (d) There is sufficient evidence at the 5% significance level to indicate that the mean pulse rate is different for women and men after doing three minutes of step-ups because 0 is not in this confidence interval. (e) None of the above

10...