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1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. If a random sample of 10 of these batteries has a variance of 1.44, which set of hypotheses is most appropriate to test the claim that the population variance is more than 0.81? 

H subscript 0 colon space mu space greater or equal than 0.81 H subscript 1 colon space mu space less than 0.81 space

 H subscript 0 colon space sigma squared space less or equal than 0.81 H subscript 1 colon space sigma squared space greater than 0.81 space

     H subscript 0 colon space sigma squared space equals 0.81 H subscript 1 colon space sigma squared space not equal to 0.81 space

H subscript 0 colon space sigma squared space greater or equal than 0.81 H subscript 1 colon space sigma squared space less than 0.81 space

10 points   

QUESTION 2

1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. A random sample of 10 of these batteries has a variance of 1.44. We are interested in testing the claim that the population variance is more than 0.81. Which test statistic is appropriate ?

top enclose X

S squared

fraction numerator left parenthesis n minus 1 right parenthesis S squared over denominator 0.81 end fraction

S

10 points   

QUESTION 3

1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. A random sample of 10 of these batteries has a variance of 1.44. You  will test the claim that the population variance is more than 0.81. Calculate the value of the test statistic. 

16

3

square root of 2

100

10 points   

QUESTION 4

1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. A random sample of 10 of these batteries has a variance of 1.44. You will test the claim that the population variance is more than 0.81. Define the rejection region for an alpha equals 0.05 level test. 

Test Statistic > X subscript 0.05 end subscript squared where X squared  has a chi-square distribution with 9 degrees of freedom.

Test Statistic < X subscript 0.05 end subscript squared where X squared  has a chi-square distribution with 9 degrees of freedom. 

Test Statistic > X subscript 0.025 end subscript squared where X squared  has a chi-square distribution with 9 degrees of freedom. 

 

Test Statistic > X subscript 0.05 end subscript squared where X squared  has a chi-square distribution with 10 degrees of freedom. 

10 points   

QUESTION 5

1. The null hypothesis should be rejected when the p-value is less than alpha where alpha is the level of significance for the test. 

 True

 False

10 points   

QUESTION 6

1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. A random sample of 10 of these batteries has a variance of 1.44. You will test the claim that the population variance is more than 0.81 using an alpha equals 0.05 level of significance. The p-value for this test is 0.07. Make an appropriate conclusion.

Reject the null hypothesis and conclude that the population variance is greater than 0.81.

Fail to reject the null hypothesis and conclude that the population variance is less than or equal to 0.81.

The test is inconclusive because the p-value is nearly 0. 

One can not make a decision with the p-value alone. We need both a p-value and the value of test statistic to make a conclusion.

10 points   

QUESTION 7

1. A manufacturer of car batteries claims that the life of the company’s batteries in years is approximately normally distributed with a population variance of 0.81. A random sample of 10 of these batteries has a variance of 1.44. Construct a 95% C.I. for sigma squared. Use  X subscript 0.975 end subscript squared equals 1.689 and X subscript 0.025 end subscript squared equals 16.013 to construct the interval. 

(-1.023,1.567)

(0.809, 7.673)

(0.678,9.081)

(0.81,1.44)

10 points   

QUESTION 8

1. An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave a sample standard deviation of 4, while the samples of material 2 gave a sample standard deviation of 5. The researcher wants to test the claim that the two material have different population variances. Determine an appropriate set of hypotheses to test this claim.

H subscript 0 colon sigma subscript 1 squared equals sigma subscript 2 squared

H subscript 1 colon sigma subscript 1 squared not equal to sigma subscript 2 squared

H subscript 0 colon sigma subscript 1 squared less or equal than sigma subscript 2 squared

H subscript 1 colon sigma subscript 1 squared greater than sigma subscript 2 squared

H subscript 0 colon sigma subscript 1 squared greater or equal than sigma subscript 2 squared

H subscript 1 colon sigma subscript 1 squared less than sigma subscript 2 squared

H subscript 0 colon sigma subscript 1 squared less or equal than sigma subscript 2 squared

H subscript 1 colon sigma subscript 1 squared greater or equal than sigma subscript 2 squared

10 points   

QUESTION 9

1. An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave a sample standard deviation of 4, while the samples of material 2 gave a sample standard deviation of 5. The researcher wants to test the claim that the two material have different population variances. Identify an appropriate test statistic.

S subscript 1 squared over S subscript 2 squared

top enclose X

S squared

S

10 points   

QUESTION 10

1. An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave a sample standard deviation of 4, while the samples of material 2 gave a sample standard deviation of 5. The researcher wants to test the claim that the two material have different population variances. Determine the rejection region for this test. Use an alpha equals 0.05 level of significance. 

Test Statistic > F subscript 0.05 end subscript where F is the F distribution with 11 numerator degrees of freedom and 9 denominator degrees of freedom. 

Test Statistic < F subscript 0.05 end subscript where F is the F distribution with 11 numerator degrees of freedom and 9 denominator degrees of freedom. 

Test Statistic < F subscript 0.975 end subscript or Test Statistic >  F subscript 0.025 end subscript where F is the F distribution with 11 numerator degrees of freedom and 9 denominator degrees of freedom. 

Test Statistic > F subscript 0.05 end subscript where F is the F distribution with 9 numerator degrees of freedom and 11 denominator degrees of freedom. 

10 points   

QUESTION 11

1. An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave a sample standard deviation of 4, while the samples of material 2 gave a sample standard deviation of 5. The researcher wants to test the claim that the two material have different population variances using an alpha equals 0.05 level of significance. The p-value for this test is 0.19. Use the this information to arrive at a conclusion.

Reject the null hypothesis and conclude that the population variances are not equal.

Reject the null hypothesis and conclude that the population variance for the first population is greater than that of the second population. 

Fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest that the population variances differ. 

The test is inconclusive because the p-value is larger than the level of significance. 

10 points   

QUESTION 12

1. During the first 13 weeks of the television season, the Saturday evening 8:00 p.m. to 9:00 p.m. audience proportions were recorded as ABC 29%, CBS 28%, NBC 25%, and independents 18%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and independents 46 homes. Test to determine whether the viewing audience proportions changed. Choose the appropriate set of hypotheses. 

H subscript 0 colon space p subscript A B C equals end subscript 0.29 comma p subscript C B S equals end subscript 0.28 comma space p subscript N B C equals end subscript 0.25 space a n d space p subscript I N D. equals end subscript 0.18

H subscript 1 colon space T h e space p o p u l a t i o n space p r o p o r t i o n s space a r e space n o t space p subscript A B C equals end subscript 0.29 comma p subscript C B S equals end subscript 0.28 comma space p subscript N B C equals end subscript 0.25 space a n d space p subscript I N D. equals end subscript 0.18

H subscript 0 colon space T h e space p o p u l a t i o n space p r o p o r t i o n s space a r e space n o t space p subscript A B C equals end subscript 0.29 comma p subscript C B S equals end subscript 0.28 comma space p subscript N B C equals end subscript 0.25 space a n d space p subscript I N D. equals end subscript 0.18

H subscript 1 colon space p subscript A B C equals end subscript 0.29 comma p subscript C B S equals end subscript 0.28 comma space p subscript N B C equals end subscript 0.25 space a n d space p subscript I N D. equals end subscript 0.18

H subscript 0 colon space mu equals 0

H subscript 1 colon space mu greater than 0

H subscript 0 colon space sigma equals 1

H subscript 1 colon space sigma not equal to 1

10 points   

QUESTION 13

1. During the first 13 weeks of the television season, the Saturday evening 8:00 p.m. to 9:00 p.m. audience proportions were recorded as ABC 29%, CBS 28%, NBC 25%, and independents 18%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and independents 46 homes. Test to determine whether the viewing audience proportions changed. Calculate the test statistic.

4.045

-1.067

17.678

6.867

10 points   

QUESTION 14

1. During the first 13 weeks of the television season, the Saturday evening 8:00 p.m. to 9:00 p.m. audience proportions were recorded as ABC 29%, CBS 28%, NBC 25%, and independents 18%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and independents 46 homes. Test to determine whether the viewing audience proportions changed. Use calculation of the test statistic, together with the correct reject region to make a conclusion. The critical value for the reject region is    X subscript 0.05 end subscript squared equals 7.815 comma space w h e r e space X squared space i s space f r o m space a space c h i minus s q u a r e d space d i s t r i b u t i o n space w i t h space 3 space d e g r e e s space o f space f r e e d o m.

  

Reject the null hypothesis and conclude the population proportions are not as defined in the null.

Fail to reject the null hypothesis and conclude there is insufficient evidence in the data to suggest the population proportions are different from those defined in the null.

The test is inconclusive as the test statistic is too large. 

The test rules that both the null and the alternative are reasonable options. It's up to the researcher to make the choice.

10 points   

QUESTION 15

1. Choose the general form for the goodness-of-fit test statistic. Note that goodness-of-fit includes the test for multinomial goodness-of-fit and the test for independence. 

S squared

fraction numerator left parenthesis n minus 1 right parenthesis S squared over denominator sigma subscript 0 squared end fraction

begin inline style sum from i equals 1 to k of end style fraction numerator left parenthesis o subscript i minus e subscript i right parenthesis squared over denominator e subscript i end fraction comma space w h e r e space o subscript i space a r e space o b s e r v e d space f r e q u e n c i e s space a n d space e subscript i space a r e space e x p e c t e d space f r e q u e n c i e s.

top enclose X

10 points   

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