Stats quiz

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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

QUESTION 2

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

QUESTION 3

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

QUESTION 4

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.

QUESTION 5

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

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

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. 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

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. 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

QUESTION 8

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. 

QUESTION 9

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.

QUESTION 10

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)

QUESTION 11

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

QUESTION 12

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

QUESTION 13

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

QUESTION 14

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. 

QUESTION 15

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. 

QUESTION 16

1. Consider a completely randomized design with three treatment groups. We reject the null hypothesis, and proceed to perform multiple comparisons. What is the Bonferroni adjusted level of significance if the original level of significance is 0.05? 

0.025

0.0167

0.001

0.005

QUESTION 17

1. Consider a randomized block design with three treatments and six blocks. Note that the number of blocks is the number of subjects in the study. What is the value of the ANOVA test statistic if treatment sum of squares is 21 and the block sum of squares is 30?

5.53

0.70

0.50

5

QUESTION 18

1. Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 5 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. What set of hypotheses is appropriate for comparing the three groups in this setting?

H subscript 0 colon space mu subscript 1 equals mu subscript 2 equals mu subscript 3 H subscript 1 colon space N o t space a l l space p o p u l a t i o n space m e a n s space a r e space e q u a l

H subscript 0 colon mu equals 0 H subscript 1 colon mu not equal to 0

H subscript 0 colon space N o t space a l l space p o p u l a t i o n space m e a n s space a r e space e q u a l H subscript 1 colon space mu subscript 1 equals mu subscript 2 equals mu subscript 3 space

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

QUESTION 19

1. Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 5 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. What is the rejection region for the ANOVA test using an arbitrary level of significance alpha.

R e j e c t space H subscript 0 space i f space T E S T space S T A T I S T I C space greater than space F subscript alpha space w h e r e space F subscript alpha space i s space b a s e d space o n space a n space F space d i s t r i b u t i o n space w i t h space k minus 1 space a n d space n minus k space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space t r e a t m e n t c a t e g o r i e s space a n d space n space i s space t h e space t o t a l space s a m p l e space s i z e. space

R e j e c t space H subscript 0 space i f space T E S T space S T A T I S T I C space less than space F subscript alpha space w h e r e space F subscript alpha space i s space b a s e d space o n space a n space F space d i s t r i b u t i o n space w i t h space k minus 1 space a n d space n space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space t r e a t m e n t c a t e g o r i e s space a n d space n space i s space t h e space t o t a l space s a m p l e space s i z e. space

R e j e c t space H subscript 0 space i f space T E S T space S T A T I S T I C space greater than space X squared subscript alpha space w h e r e space X squared subscript alpha space i s space b a s e d space o n space a space X squared space d i s t r i b u t i o n space w i t h space k minus 1 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space t r e a t m e n t space g r o u p s. space

R e j e c t space H subscript 0 space i f space T E S T space S T A T I S T I C space less than space X squared subscript alpha space w h e r e space X squared subscript alpha space i s space b a s e d space o n space a space X squared space d i s t r i b u t i o n space w i t h space k minus 1 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space t r e a t m e n t space g r o u p s. space

QUESTION 20

1. Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 5 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. Use the information in the ANOVA table to calculate the test statistic for the ANOVA test. 

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260

520

28.33

9.18

QUESTION 21

1. Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 5 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. The p-value for the ANOVA test is 0.004. Choose an appropriate conclusion. Use a 0.05 level of significance.

Reject the null hypothesis and proceed to perform multiple comparisons.

Fail to reject the null and proceed to perform multiple comparisons.

Fail to reject the null and do not perform multiple comparisons. 

Reject the null and do not perform multiple comparisons.  

QUESTION 22

1. Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 5 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. A table of means as well as the ANOVA table is provided below. Use the t-value of 1.19 to carry out the comparisons. Which pairs of means are significantly different? 

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Group 1 and Group 2 ; Group 1 and Group 3

Group 1 and Group 3 ; Group 2 and Group 3 

Group 1 and Group 2 

All pairs of groups are significant

QUESTION 23

1. What is the ANOVA test statistic for a completely randomized design?

fraction numerator begin display style bevelled fraction numerator S S T over denominator k minus 1 end fraction end style over denominator begin display style bevelled fraction numerator S S E over denominator n minus k end fraction end style end fraction space w h e r e space k space i s space t h e space n u m b e r space o f space t r e a t m e n t space a n d space n space i s space t h e space t o t a l space s a m p l e space s i z e.

   fraction numerator begin display style S S T end style over denominator begin display style S S E end style end fraction space

top enclose X

fraction numerator top enclose X over denominator S squared end fraction

QUESTION 24

1. Consider the goodness-of-fit test for a Poisson or Normal distribution. When the expected frequency in some category is less than 5, it is recommended that adjacent categories be combined to obtain expected frequencies that are all greater than 5. What is the reason for this recommendation?

It makes the calculation of the test statistic easier.

To avoid division by zero when calculating the test statistic.

To avoid an overflow error when using software to calculate the test statistic.

The assumptions for the goodness-of-fit test are not satisfied when there is an expected frequency that is less than 5. 

QUESTION 25

1. Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What hypotheses are appropriate for this test?

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H subscript 0 colon space T h e space p o p u l a t i o n space h a s space a space N o r m a l space d i s t r i b u t i o n H subscript 1 colon space T h e space p o p u l a t i o n space d o e s space n o t space h a v e space a space N o r m a l space d i s t r i b u t i o n

H subscript 0 colon space T h e space p o p u l a t i o n space d o e s space n o t space h a v e space a space N o r m a l space d i s t r i b u t i o n H subscript 1 colon space T h e space p o p u l a t i o n space d o e s space h a v e space a space N o r m a l space D i s t r i b u t i o n

H subscript 0 colon space T h e space p o p u l a t i o n space d o e s space n o t space h a v e space a space P o i s s o n space d i s t r i b u t i o n H subscript 1 colon space T h e space p o p u l a t i o n space h a s space a space P o i s s o n space d i s t r i b u t i o n

H subscript 0 colon space T h e space p o p u l a t i o n space h a s space a space P o i s s o n space d i s t r i b u t i o n H subscript 1 colon space T h e space p o p u l a t i o n space d o e s space n o t space h a v e space a space P o i s s o n space d i s t r i b u t i o n

QUESTION 26

1. Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What test statistic is appropriate for this test?

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X squared equals begin inline style sum from i equals 1 to k of end style fraction numerator left parenthesis f subscript i minus e subscript i right parenthesis squared over denominator e subscript i end fraction

f left parenthesis x right parenthesis equals fraction numerator mu to the power of x e to the power of negative mu end exponent over denominator x factorial end fraction

top enclose X

S squared

QUESTION 27

1. Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What is the rejection for this test?

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R e j e c t space H subscript 0 space i f space T e s t space S t a t i s t i c less than X squared subscript alpha divided by 2 end subscript space w h e r e space alpha space i s space t h e space l e v e l space o f space s i g n i f i c a n c e a n d space t h e r e space a r e space k minus 1 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space c a t e g o r i e s.

R e j e c t space H subscript 0 space i f space T e s t space S t a t i s t i c greater than X squared subscript alpha space w h e r e space alpha space i s space t h e space l e v e l space o f space s i g n i f i c a n c e a n d space t h e r e space a r e space k minus 1 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space c a t e g o r i e s.

R e j e c t space H subscript 0 space i f space T e s t space S t a t i s t i c greater than X squared subscript alpha space w h e r e space alpha space i s space t h e space l e v e l space o f space s i g n i f i c a n c e a n d space t h e r e space a r e space k minus 2 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space c a t e g o r i e s.

R e j e c t space H subscript 0 space i f space T e s t space S t a t i s t i c less than X squared subscript alpha divided by 2 end subscript space w h e r e space alpha space i s space t h e space l e v e l space o f space s i g n i f i c a n c e a n d space t h e r e space a r e space k minus 2 space d e g r e e s space o f space f r e e d o m space w h e r e space k space i s space t h e space n u m b e r space o f space c a t e g o r i e s.

QUESTION 28

1. Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. Calculate the the value of test statistic. Use the "Number of Occurences" as the categories.

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9.042

1.30

3.142

5

QUESTION 29

1. The critical value for a 0.05 level of significance goodness-of-fit test for the Poisson distribution is the same as that for the Normal distribution.

 True

 False

QUESTION 30

1. Consider a test of H subscript 0 colon beta subscript 1 equals 0 space v s space H subscript 1 colon beta subscript 1 not equal to 0 for the simple linear regression model. The estimated regression line is 60+5x and the SSE is 1,530. The data used to build the estimated line is given below. Calculate the test statistic.

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8.62

60

1,530

5