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 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
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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
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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 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
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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
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10 points
QUESTION 2
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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S squared
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top enclose X
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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
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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
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10 points
QUESTION 3
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 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.
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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 greater 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.
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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 greater 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.
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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 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.
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10 points
QUESTION 4
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.
/var/folders/0b/_f26rqjj5x34j6z79w8j15xh0000gn/T/com.microsoft.Word/WebArchiveCopyPasteTempFiles/xid-75777851_1
10 points
QUESTION 5
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?
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It makes the calculation of the test statistic easier.
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To avoid division by zero when calculating the test statistic.
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To avoid an overflow error when using software to calculate the test statistic.
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The assumptions for the goodness-of-fit test are not satisfied when there is an expected frequency that is less than 5.
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10 points
QUESTION 6
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
10 points
QUESTION 7
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?
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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
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H subscript 0 colon sigma squared equals 1 H subscript 1 colon sigma squared not equal to 1
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H subscript 0 colon mu equals 0 H subscript 1 colon mu not equal to 0
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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
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10 points
QUESTION 8
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
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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 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 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
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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 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
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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 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
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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 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
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10 points
QUESTION 9
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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10 points
QUESTION 10
1. What is the ANOVA test statistic for a completely randomized design?
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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.
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fraction numerator begin display style S S T end style over denominator begin display style S S E end style end fraction space
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top enclose X
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fraction numerator top enclose X over denominator S squared end fraction
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10 points
QUESTION 11
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.
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Reject the null hypothesis and proceed to perform multiple comparisons.
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Fail to reject the null and proceed to perform multiple comparisons.
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Fail to reject the null and do not perform multiple comparisons.
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Reject the null and do not perform multiple comparisons.
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10 points
QUESTION 12
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
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Group 1 and Group 3 ; Group 2 and Group 3
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Group 1 and Group 2
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All pairs of groups are significant
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10 points
QUESTION 13
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?
10 points
QUESTION 14
1. Consider a randomized block design with three treatments and six blocks. What is the value of the ANOVA test statistic if treatment sum of squares is 21 and the block sum of squares is 30?
10 points
QUESTION 15
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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10 points