Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results._ \[\begin{array} { c | r | r | r | r | r | r } \text { Number } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Frequency } & 4 & 13 & 2 & 14 & 13 & 2 \end{array}\] Use a significance level of 0.05 to test the claim that the die is fair.

H₀: The die is fair (all numbers occur w

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the flu was recorded. The results are shown below. \[\begin{array} { r | c c c } & \text { Vaccinated } & \text { Placebo } & \text { Control } \\ \hline \text { Caught the flu } & 8 & 19 & 21 \\ \text { Did not catch the flu } & 142 & 161 & 79 \end{array}\] Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

@#LAT-DLM& : The proportion of people catching the

Use the sample data below to test whether car color affects the likelihood of being in an_ accident. Use a significance level of 0.01. \[\begin{array} { c | c c c } & \text { Red } & \text { Blue } & \text { White } \\ \hline \text { Car has been in accident } & 28 & 33 & 36 \\ \text { Car has not been in accident } & 23 & 22 & 30 \end{array}\]

@#LAT-DLM& Car color and being in an accident are

Exam 11: Chi-Square and Analysis of Variance

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of significance that success is independent of the method used, technology provides a P-value of 0.0355. What does the P-value tell us about the claim?

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Question 1

Correct Answer:

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Since the P-value of 0.0355 is lower than 0.05, we reject the null hypothesis of independence between the
treatment and whether the subject stops experiencing back pain. This suggests that the choice of treatment does
appear to make a difference.

According to Benford's Law, a variety of different data sets include numbers with leading (first)digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford's law: distribution of leading digits 30.1\% 17.6\% 12.5\% 9.7\% 7.9\% 6.7\% 5.8\% 5.1\% 4.6\% When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476, 180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's Law. Does it appear that the checks are the result of fraud?

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Question 2

Correct Answer:

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Test statistic $\chi ^ { 2 } = 3590.094$ . Critical value: $\chi ^ { 2 } = 15.507 . P$ -value $< 0.005$ . There is sufficient evidence to warrant rejection of the claim that the leading digits are from a population with a distribution that conforms to Benford's Law. It does appear that the checks are the result of fraud.

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and asked whether they planned to vote in the next election. The results are shown below. Men Women Plan to vote 170 185 Do not plan to vote 130 115

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Question 3

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$H _ { 0 }$ : The proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote.
$H _ { 1 }$ : At least one of the proportions are different.
Test statistic: $\chi ^ { 2 } = 1.552$ . Critical value: $\chi ^ { 2 } = 6.635$ .
Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote.

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of significance that success is independent of the method used, technology provides a P-value of 0.0655. What does the P-value tell us about the claim?

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Question 4

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results._ Number 1 2 3 4 5 6 Frequency 4 13 2 14 13 2 Use a significance level of 0.05 to test the claim that the die is fair.

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Question 5

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and asked whether they planned to vote in the next election. The results are shown below. Men Women Plan to vote 170 185 Do not plan to vote 130 115

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Question 6

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the flu was recorded. The results are shown below. Vaccinated Placebo Control Caught the flu 8 19 21 Did not catch the flu 142 161 79 Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

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Question 7

Use the sample data below to test whether car color affects the likelihood of being in an_ accident. Use a significance level of 0.01. Red Blue White Car has been in accident 28 33 36 Car has not been in accident 23 22 30

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Question 8

Among the four northwestern states, Washington has 51% of the total population, Oregon has 30%, Idaho has 11%, and Montana has 8%. A market researcher selects a sample of 1000 subjects, with 450 in Washington, 340 in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance level, test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state populations.

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Question 9

Explain the computation of expected values for contingency tables in terms of probabilities._ Refer to the assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

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Question 10

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of Significance that success is independent of the method used, technology provides a P-value Of 0.0355. What does the P-value tell us about the claim?

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Question 11

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the flu was recorded. The results are shown below. Vaccinated Placebo Control Caught the flu 8 19 21 Did not catch the flu 142 161 79

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Question 12

Which of the following is not a characteristic of a chi-square distribution?

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Question 13

Use a $X^2$ test to test the claim that in the given contingency table, the row variable and the_ column variable are independent. 160 students who were majoring in either math or English were asked a test question, and the researcher recorded whether they answered the question correctly. The sample results are given below. At the 0.10 significance level, test the claim that response and major are independent. Correct Incorrect Math 27 53 English 43 37

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Question 14

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or owned their own home. The results are shown in the table below sorted by gender. At $\alpha$ = 0.05, test the claim that living accommodations are independent of the gender of the student. Live with Parent Rent Apartment Own Home Male 20 26 19 Female 18 28 30

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Question 15

Use a $X^2$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. Responses to a survey question are broken down according to employment status and the sample results are given below. At the 0.10 significance level, test the claim that response and employment status are independent. Yes No Undecided Employed 30 15 5 Unemployed 20 25 10

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Question 16

Describe the null hypothesis for the test of independence. List the assumptions for the $X^2$ test_ of independence.

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Question 17

According to Benford's Law, a variety of different data sets include numbers with leading (first)7)____________ digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford's law: distribution of leading digits 30.1\% 17.6\% 12.5\% 9.7\% 7.9\% 6.7\% 5.8\% 5.1\% 4.6\% When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476, 180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's Law. Does it appear that the checks are the result of fraud?

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Question 18

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

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Question 19

Use a $X^2$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. 160 students who were majoring in either math or English were asked a test question, and the researcher recorded whether they answered the question correctly. The sample results are given below. At the 0.10 significance level, test the claim that response and major are independent. Correct Incorrect Math 27 53 English 43 37

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Question 20

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results._ Number 1 2 3 4 5 6 Frequency 4 13 2 14 13 2 Use a significance level of 0.05 to test the claim that the die is fair.

Use the sample data below to test whether car color affects the likelihood of being in an_ accident. Use a significance level of 0.01. Red Blue White Car has been in accident 28 33 36 Car has not been in accident 23 22 30

Explain the computation of expected values for contingency tables in terms of probabilities._ Refer to the assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

Which of the following is not a characteristic of a chi-square distribution?

Describe the null hypothesis for the test of independence. List the assumptions for the $X^2$ test_ of independence.

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Introduction to Statistics

Exploring Data With Tables and Graphs

Describing, Exploring, and Comparing Data

Probability

Discrete Probability Distributions

Normal Probability Distributions

Estimating Parameters and Determining Sample Sizes

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

Statistical Control Charts, Nonparametric Tests, and Hypothesis Testing

Filters

Exam 11: Chi-Square and Analysis of Variance

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results._ Number 1 2 3 4 5 6 Frequency 4 13 2 14 13 2 Use a significance level of 0.05 to test the claim that the die is fair.

Use the sample data below to test whether car color affects the likelihood of being in an_ accident. Use a significance level of 0.01. Red Blue White Car has been in accident 28 33 36 Car has not been in accident 23 22 30

Explain the computation of expected values for contingency tables in terms of probabilities._ Refer to the assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

Which of the following is not a characteristic of a chi-square distribution?

Describe the null hypothesis for the test of independence. List the assumptions for the X2X^2X2 test_ of independence.

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Introduction to Statistics

Exploring Data With Tables and Graphs

Describing, Exploring, and Comparing Data

Probability

Discrete Probability Distributions

Normal Probability Distributions

Estimating Parameters and Determining Sample Sizes

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

Statistical Control Charts, Nonparametric Tests, and Hypothesis Testing

Filters

Describe the null hypothesis for the test of independence. List the assumptions for the $X^2$ test_ of independence.