Question 1. In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. If we want to use the methods for testing a claim about two population proportions to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.
Choose the correct answer below.
A. The requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the sample size is at least 1000.
B. The requirements are not satisfied; the difference between the rates of those that developed polio in the two groups is not statistically significant.
C. The requirements are not satisfied; the samples are not simple random samples that are independent.
D. The requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the number of successes is at least 5 and the number of failures is at least 5.
Question 2. In a study of treatments for very painful "cluster" headaches, 148 patients were treated with oxygen and 146 other patients were given a placebo consisting of ordinary air. Among the 148 patients in the oxygen treatment group, 112 were free from headaches 15 minutes after treatment. Among the 146 patients given the placebo, 20 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hypothesis test?
A. H0: p1 = p2
H1: p1 > p2
B. H0: p1 ≤ p2
H1: p1 ≠ p2
C. H0: p1 = p2
H1: p1 < p2
D. H0: p1 ≥ p2
H1: p1 ≠ p2
E. H0: p1 = p2
H1: p1 ≠ p2
F. H0: p1 ≠ p2
H1: p1 = p2
Identify the test statics.
Identify the P-value.
What is the conclusion based on the hypothesis test?
b. Test the claim by constructing an appropriate confidence interval.
What is the conclusion based on the confidence interval?
c. Based on the results, is the oxygen treatment effective?
A. The results suggest that the oxygen treatment is not effective in curing "cluster" headaches because the cure rate for the oxygen treatment appears to be lower than that of the placebo.
B. The results suggest that the oxygen treatment is not effective in curing "cluster" headaches because the cure rates appear to be the same.
C. The results suggest that the Oxygen treatment is effective in curing "cluster" headaches.
D. The results are inconclusive.
Question 3. In a random samples of males, it was found that 24 write with their left hands and 218 do not. In a random sample of females, it was found that 68 write with their left hands and 442 do not. Use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of males and the second sample of females. What are the null and alternative hypotheses for the hypothesis test?
A. H0: p1 ≥ p2
H1: p1 ≠ p2
B. H0: p1 = p2
H1: p1 ≠ p2
C. H0: p1 ≠ p2
H1: p1 = p2
D. H0: p1 = p2
H1: p1 > p2
E. H0: p1 = p2
H1: p1 < p2
F. H0: p1 ≤ p2
H1: p1 ≠ p2
Identify the test statics.
Identify the P-value.
What is the conclusion based on the hypothesis test?
b. Test the claim by constructing an appropriate confidence interval.
What is the conclusion based on the confidence interval?
c. Based on the results, is the rate of left-handedness among less than the rate of left-handedness among females?
A. The rate of left-handedness among males does not appear to be less than the rate of left-handedness among females.
B. The rate of left-handedness among males does appear to be less than the rate of left-handedness among females because the results are statistically significant.
C. The rate of left-handedness among males does appear to be less than the rate of left-handedness among females because the results are not statistically significant.
D. The results are inconclusive
Question 4. Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea 43 of the 48 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 75 of the 91 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of subjects treated with echinacea and the second sample to be the sample of subjects treated with a placebo. What are the null and alternative hypotheses for the hypothesis test?
A. H0: p1 ≠ p2
H1: p1 = p2
B. H0: p1 ≥ p2
H1: p1 ≠ p2
C. H0: p1 ≤ p2
H1: p1 ≠ p2
D. H0: p1 = p2
H1: p1 < p2
E. H0: p1 = p2
H1: p1 > p2
F. H0: p1 = p2
H1: p1 ≠ p2
Identify the test statics.
Identify the P-value.
What is the conclusion based on the hypothesis test?
b. Test the claim by constructing an appropriate confidence interval.
What is the conclusion based on the confidence interval?
c. Based on the results, does echinacea appear to have any effect on the infection rate?
A. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it lowers the infection rate.
B. Echinacea does appear to have a significant effect on the infection rate. There is evidence that it increases the infection rate.
C. Echinacea does not appear to have a significant effect on the infection rate.
D. The results are inconclusive.
Question 5. Determine whether the samples are independent or dependent.
To test the effectiveness of a drug comma cholesterol levels are measured in 250 men beforeTo test the effectiveness of a drug, cholesterol levels are measured in 250 men before and after the treatment.and after the treatment.
Choose the correct answer below.
A. The samples are independent because there is not a natural pairing between the two samples.
B. The samples are dependent because there is a natural pairing between the two samples.
C. The samples are independent because there is a natural pairing between the two samples.
D. The samples are dependent because there is not a natural pairing between the two samples.
Question 6. Researchers conducted a study to determine whether magnets are effective in treating back pain. The results are shown in the table for the treatment (with magnets) group and the sham (or placebo) group. The results are measures of reduction in back pain. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use 0.05 significance level for both parts.
|
|
Treatment
|
Sham
|
|
μ
|
μ1
|
μ2
|
|
n
|
24
|
24
|
|
x
|
0.52
|
0.42
|
|
s
|
0.55
|
1.12
|
a. Test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment. What are the null and alternative hypotheses?
A. H0: μ1 < μ2
H1: μ1 ≥ μ2
B. H0: μ1 = μ2
H1: μ1 ≠ μ2
C. H0: μ1 ≠ μ2
H1: μ1 < μ2
D. H0: μ1 = μ2
H1: μ1 > μ2
Identify the test statics.
Identify the P-value.
State the conclusion for the test.
Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?
b. Construct a confidence interval suitable testing the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.
Question 7. A study was done using a treatment group and a placebo group. The results are shorn in the table. Assume that the two samples are independent simple random samples selected from normally distributed population, and do not assume that the population standard deviation are equal. Complete parts (a) and (b) below. Use 0.10 significance level for both parts.
|
|
Treatment
|
Placebo
|
|
μ
|
μ1
|
μ2
|
|
n
|
25
|
30
|
|
x
|
2.34
|
2.67
|
|
s
|
0.98
|
0.64
|
a. Test the claim that the two samples are from population with the same mean. What are the null and alternative hypotheses?
A. H0: μ1 = μ2
H1: μ1 > μ2
B. H0: μ1 = μ2
H1: μ1 ≠ μ2
C. H0: μ1 < μ2
H1: μ1 ≥ μ2
D. H0: μ1 ≠ μ2
H1: μ1 < μ2
Identify the test statics.
Identify the P-value.
State the conclusion for the test.
A. Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
B. Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
C. Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
D. Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
b. Construct a confidence interval suitable for testing the claim that the two samples are from populations with the same mean.
Question 8. Listed in the data table are IQ scores for a random sample of subjects with median lead levels in their blood. Also listed are statistics from a study done of IQ scores for a random sample of subjects with high lead levels. Assume that the samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.05 significance level for both parts.
a. Test the claim that the mean IQ scores for subjects with medium lead levels is higher than the mean for subjects with high lead levels.
What are the null and alternative hypotheses? Assume that population 1 consists of subjects with medium lead levels and population 2 consists of subjects with high lead levels.
A. H0: μ1 ≤ μ2
H1: μ1 > μ2
B. H0: μ1 = μ2
H1: μ1 > μ2
C. H0: μ1 ≠ μ2
H1: μ1 > μ2
D. H0: μ1 = μ2
H1: μ1 ≠ μ2
Identify the test statics.
Identify the P-value.
State the conclusion for the test.
A. Fail to reject the null hypothesis. There is sufficient evidence to support the claim that subjects with medium lead levels have higher IQ scores
B. Reject the null hypothesis. There is not sufficient evidence to support the claim that subjects with medium lead levels have higher IQ scores
C. Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that subjects with medium lead levels have higher IQ scores
D. Reject the null hypothesis. There is sufficient evidence to support the claim that subjects with medium lead levels have higher IQ scores
b. Construct a confidence interval suitable for testing the claim that the mean IQ scores for subjects with medium lead levels is higher than the mean for subjects with high lead levels
Does the confidence interval support the conclusion of the test?
Question 9. Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)? Select all that apply.
A. If one has fifteen matched pairs of sample data, there is a loose requirement that the fifteen differences appear to be from a normally distributed population.
B. The methods used to evaluate the mean of the differences between two dependent variables apply if one has 63 IQ scores of men from Ohio and 63 IQ scores of men from California
C. If one has more than 22 matched pairs of sample data, one can consider the sample to be large and there is no need to check for normality
D. The requirement of a simple random sample is satisfied if we have dependent pairs of convenience sampling data
E. If one wants to use a confidence interval to test the claim that μd > 0 with a 0.05 significance level, the confidence interval should have a confidence level of 90%.
Question 10. Data on the numbers of hospital admissions resulting from motor vehicle crashes are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the mean of the population of differences between hospital admissions. Use the confidence interval to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.
|
Friday the 6th
|
10
|
2
|
4
|
7
|
10
|
|
Friday the 13th
|
14
|
14
|
14
|
13
|
13
|
In this example, μd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the number of hospital admissions on Friday the 6th minus the number of hospital admissions on Friday the 13th. Find the 95% confidence interval.
Based on the confidence interval, can one reject the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected?
A. No, because the confidence 'atonal includes zero.
B. Yes, because the confidence Interval includes zero.
C. No, because the confidence interval does not include zero.
D. Yes, because the confidence interval does not include zero.
Question 11. Listed below are systolic blood pressure measurements (mm Hg) taken from the right and left arms of the same woman. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Use a 0.01 significance level to test for a difference between the measurements from the two arms. What can be concluded?
|
Right arm
|
142
|
150
|
122
|
130
|
129
|
|
Left arm
|
180
|
174
|
183
|
156
|
144
|
In this example, μd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the measurement from the right arm minus the measurement from the left arm. What are the null and alternative hypotheses for the hypothesis test?
A. H0: μd = 0
H1: μd < 0
B. H0: μd = 0
H1: μd ≠ 0
C. H0: μd ≠ 0
H1: μd > 0
D. H0: μd ≠ 0
H1: μd = 0
Identify the test statics.
Identify the P-value.
What is the conclusion based on the hypothesis test?
Question 12. The following data lists the ages of a random selection of actresses when they won an award in the category of Best Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete pans (a) and (b) below.
|
Actress (years)
|
30
|
28
|
32
|
29
|
35
|
26
|
25
|
39
|
28
|
31
|
|
Actor (years)
|
66
|
41
|
35
|
42
|
33
|
34
|
46
|
36
|
33
|
38
|
a. Use the sample data with a 0 01 significance level to test the claim that for the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0 (indicating that the Best Actresses are generally younger than Best Actors)
In this example, μd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the actress's age minus the actor's age. What are the null and alternative hypotheses for the hypothesis test?
Identify the test statistic
Identify the P-value
What is the conclusion based on the hypothesis test?
b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
What feature of the confidence interval leads to the same conclusion reached in part (a)?