## (solution) 3. Conduct a test at the = 0.10 level of significance by

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3. Conduct a test at the = 0.10 level of significance by determining (a) the null and alternative hypotheses, (b) the test

statistic, and (c) the P­value. Assume the samples were obtained independently from a large population using simple random sampling.

Test whether p1 &gt; p2 . The sample data are x1 = 125, n1 = 243, x2 = 144, and n2 = 305.

(a) Choose the correct null and alternative hypotheses below.

A. H0 : p1 = p2 versus H1 : p1 &lt; p2

B. H0 : p1 = 0 versus H1 : p1 ? 0

C. H0 : p1 = p2 versus H1 : p1 &gt; p2

D. H0 : p1 = p2 versus H1 : p1 ? p2

(b) Determine the test statistic.

z0 = (Round to two decimal places as needed.) (c) Determine the P­value.

The P­value is .

(Round to three decimal places as needed.)

What is the result of this hypothesis test?

A. Do not reject the null hypothesis because there is not sufficient evidence to conclude that p1 ? p2 .

B. Do not reject the null hypothesis because there is not sufficient evidence to conclude that p1 &gt; p2 .

C. Do not reject the null hypothesis because there is not sufficient evidence to conclude that p1 &lt; p2 .

D. Reject the null hypothesis because there is sufficient evidence to conclude that p1 &lt; p2 .

4. Construct a confidence interval for p1 ? p2 at the given level of confidence.

x1 = 399, n1 = 532, x2 = 423, n2 = 598, 95% confidence

The 95% confidence interval for p1 ? p2 is ( , (Use ascending order. Round to three decimal places as needed.) ). 5. In a clinical trial of a vaccine, 8,000 children were randomly divided into two groups. The subjects in group 1 (the experimental group) were given the vaccine while the subjects in group 2 (the control group) were given a placebo. Of the 4,000 children in the experimental group, 73 developed the disease. Of the 4,000 children in the control group, 106 developed the disease.

Determine whether the proportion of subjects in the experimental group who contracted the disease is less than the proportion of subjects in the control group who contracted the disease at the = 0.01 level of significance. Choose the correct null and alternative hypotheses below.

A. H0 : p1 = p2 versus H1 : p1 &gt; p2

B. H0 : p1 = 0 versus H1 : p1 &lt; 0

C. H0 : p1 = p2 versus H1 : p1 &lt; p2

D. H0 : p1 = p2 versus H1 : p1 ? p2

Determine the test statistic.

z0 = (Round to two decimal places as needed.) Determine the P­value.

The P­value is . (Round to three decimal places as needed.) What is the result of this hypothesis test?

A. Do not reject the null hypothesis because there is sufficient evidence to conclude that the proportion of subjects in the experimental group who contracted the disease is less than the proportion of subjects in the control group at = 0.01.

B. Do not reject the null hypothesis because there is not sufficient evidence to conclude that the proportion of subjects in the experimental group who contracted the disease is less than the proportion of subjects in the control group at = 0.01.

C. Reject the null hypothesis because there is not sufficient evidence to conclude that the proportion of subjects in the experimental group who contracted the disease is less than the proportion of subjects in the control group at = 0.01.

D. Reject the null hypothesis because there is sufficient evidence to conclude that the proportion of subjects in the experimental group who contracted the disease is less than the proportion of subjects in the control group at = 0.01.

6. Fill in the blank below.

A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched­pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi, and di = Xi ? Yi, then the null hypothesis is H0: d = 0 and the alternative hypothesis is H1: d ___ 0.

A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched­pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi, and di = Xi ? Yi, then the null hypothesis is H0: d = 0 and the alternative hypothesis is H1: d &lt; 0. 7. The following data represent the muzzle velocity (in feet per second) of shells fired from a 155­mm gun. For each shell, two measurements of the velocity were recorded using two different measuring devices, resulting in the following data.

Observation

A

B 1

793.2

802.8 2

792.6

796.8 3

791.4

790.2 4

790.5

792.8 5

790.7

786.1 6 792.5

789.0 (a) Why are these matched­pairs data?

A. The measurements (A and B) are taken by the same instrument.

B. All the measurements came from rounds fired from the same gun.

C. The same round was fired in every trial.

D. Two measurements (A and B) are taken on the same round.

(b) Is there a difference in the measurement of the muzzle velocity between device A and device B at the = 0.01 level of

significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. What is your conclusion regarding H0 ?

Reject H0 .

Do not reject H0 .

(c) Construct a 99% confidence interval about the population mean difference. Compute the difference as device A minus device B. Interpret your results.

The confidence interval is (

,

(Round to three decimal places as needed.) ). Choose the statement that best agrees with your interpretion of your results.

A. I am 99% confident that the mean difference in measurement lies in the interval found above.

B. I am 18% confident that the mean difference in measurement is 0.

C. I am 18% confident that the mean difference in measurement is 0.01.

D. I am 1% confident that the mean difference in measurement lies in the interval found above. 8. A researcher studies water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The researcher immerses a weighted disk painted black and white and measures the depth (in inches) at which it is no longer visible. The collected data is given in the table below. Complete parts (a) and (b) below.

Observation

Date 1

1/25 2

3/19 3

5/30 4

7/3 5

9/13 6 11/7 Initial

After five years 55.7

52.0 47.3

53.0 42.5

41.8 41.1

41.3 56.1

58.3 60.7

61.6 (a) Why is it important to take the measurements on the same date?

A. Using the same dates maximizes the difference in water clarity.

B. Those are the same dates that all biologists use to take water clarity samples.

C. Using the same dates makes the second sample dependent on the first.

D. Using the same dates makes it easier to remember to take samples.

(b) Does the evidence suggest that the clarity of the lake is improving at the = 0.05 level of significance? Note that the normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Choose the correct conclusion below.

Reject H0 .

Do not reject H0 .

9. A researcher randomly selects 6 fathers who have adult sons and records the fathers' and sons' heights to obtain the data below. Determine if sons are taller than their fathers at the = 0.1 level of significance. The normal probability plot and boxplot indicate that the differences are approximately normally distributed with no outliers.

Observation

Height of father (in inches)

Height of son (in inches) 1

2

3

4

5

6 72.3 67.9 72.4 72.9 71.5 74.8

71.7 66.4 74.3 73.3 69.9 75.9 Using the differences (father's height) ? (son's height), what is your conclusion regarding H0 ?

Reject H0 because t0 is less than ? t .

Do not reject H0 because t0 is less than ? t .

Do not reject H0 because t0 is greater than ? t .

Reject H0 because t0 is greater than ? t .

10. Assuming that both populations are normally distributed, construct a 90% confidence interval about 1 ? 2 . ( 1 represents the mean of the experimental group and 2 represents the mean of the control group.) n

x

s The confidence interval has a lower bound of and an upper bound of (Use ascending order. Round to two decimal places as needed.) Experimental

22

45.1

5.9

. Control

20

47.9

9.8 11. Test whether 1 &lt; 2 at the = 0.05 level of significance for the sample data shown in the accompanying table. Assume that the populations are normally distributed.

1 Click the icon to view the data table.

Determine the null and alternative hypothesis for this test. A. H0 : 1 ? 2

H1 : 1 &lt; 2

B. H0 : 1 = 2

H1 : 1 &lt; 2

C. H0 : 1 = 2

H1 : 1 ? 2

D. H0 : 1 &lt; 2

H1 : 1 = 2

Detemine the P­value for this hypothesis test. P= (Round to three decimal places as needed.) State the appropriate conclusion. Choose the correct answer below.

A. Do not reject H0 . There is sufficient evidence at the = 0.05 level of significance to conclude that 1 &lt; 2. B. Reject H0 . There is sufficient evidence at the = 0.05 level of significance to conclude that 1 &lt; 2 .

C. Reject H0 . There is not sufficient evidence at the = 0.05 level of significance to conclude that 1 &lt; 2 .

D. Do not reject H0 . There is not sufficient evidence at the = 0.05 level of significance to conclude that 1 &lt; 2. 1: Sample Data

Population 1 Population 2 n 31 25 x 103.4 114.2 s 12.3 13.3 12. In baseball, league A allows a designated hitter (DH) to bat for the pitcher, who is typically a weak hitter. In league B, the pitcher must bat. The common belief is that this results in league A teams scoring more runs. In interleague play, when league A teams visit league B teams, the league A pitcher must bat. So, if the DH does result in more runs, it would be expected that league A teams will score fewer runs when visiting league B parks. To test this claim, a random sample of runs scored by league A teams with and without their DH is given in the accompanying table. Does the designated hitter result in more runs scored at the = 0.05 level of significance? Note that xA = 6.0, sA = 3.5, xB = 4.3, and sB = 2.6.

2 Click the icon to view the data table. Determine the null and alternative hypotheses for this test. A. H0 : A = B

H1 : A &gt; B

B. H0 : A = B

H1 : A &lt; B

C. H0 : A &gt; B

H1 : A = B

D. H0 : A = B

H1 : A ? B

Determine the P­value for this test.

P­value = (Round to three decimal places as needed.) State the appropriate conclusion. Choose the correct answer below.

A. Do not reject H0 . There is not sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

B. Reject H0 . There is not sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

C. Reject H0 . There is sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

D. Do not reject H0 . There is sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs. 2: Sample of Runs

Full data set League A Park (with DH)

6 2 3 6 8 2 3 7 6 4 4 12 5 6 13 6 9 5 6 7 4 3 2 5 5 6 14 14 7 0 League B Park (without DH)

1 5 5 4 7 1 6 2 9 2 8 8 2 10 4 4 3 4 1 9 3 5 1 3 3 3 5 2 7 2 13. A random sample of 40 adults with no children under the age of 18 years results in a mean daily leisure time of 5.39 hours, with a standard deviation of 2.33 hours. A random sample of 40 adults with children under the age of 18 results in a mean daily leisure time of 4.39 hours, with a standard deviation of 1.77 hours. Construct and interpret a 95% confidence interval for the mean difference in leisure time between adults with no children and adults with children ( 1 ? 2 ).

Let 1 represent the mean leisure hours of adults with no children under the age of 18 and 2 represent the mean leisure hours of adults with children under the age of 18.

The 95% confidence interval for ( 1 ? 2 ) is the range from (Round to two decimal places as needed.) hours to hours. What is the interpretation of this confidence interval?

A. There is a 95% probability that the difference of the means is in the interval. Conclude that there is insufficient evidence of a significant difference in the number of leisure hours.

B. There is a 95% probability that the difference of the means is in the interval. Conclude that there is a significant difference in the number of leisure hours.

C. There is 95% confidence that the difference of the means is in the interval. Conclude that there is a significant difference in the number of leisure hours.

D. There is 95% confidence that the difference of the means is in the interval. Conclude that there is insufficient evidence of a significant difference in the number of leisure hours.

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