At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results have been obtained. $\begin{array} { l } \text { Brand A}&\text { Brand B}&\text { Brand C }&\text { Brand D }\\\hline 17&18&21&22\\20&18&24&25\\21&23&25&27\\22&25&26&29\\21&26&29&35\\&&29&36\\&&&37 \end{array}$

@#LAT-DLM& The means are not all equal. @#LAT-DLM&-value: @#LAT-DLM&.

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -Given the sample data below, test the claim that the populations have the same mean. Use a significance level of 0.05. $\begin{array} { l } \text { Brand A }&\text { Brand B }&\text { Brand C }&\text { Brand D }\\\hline \mathrm{n}=16 & \mathrm{n}=16 & \mathrm{n}=16 & \mathrm{n}=16 \\ \overline{\mathrm{x}}=2.09 & \overline{\mathrm{x}}=3.48 & \overline{\mathrm{x}}=1.86 & \overline{\mathrm{x}}=2.84 \\ \mathrm{~s}=0.37 & \mathrm{~s}=0.61 & \mathrm{~s}=0.45 & \mathrm{~s}=0.53 \end{array}$

Test statistic: @#LAT-DLM&. Critical val

What can you conclude about the equality of the population means? $\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\ \text { Error } & 16 & 13.925 & 0.870 & & \\ \text { Total } & 19 & 27.425 & & & \end{array}$

A) Accept the null hypothesis since the p-value is greater than the significance level. B) Reject the null hypothesis since the p-value is less than the significance level. C) Reject the null hypothesis since the p-value is greater than the significance level. D) Accept the null hypothesis since the p-value is less than the significance level. A) Accept the null hypothesis since the p-value is greater than the significance level. B) Reject the null hypothesis since the p-value is less than the significance level. C) Reject the null hypothesis since the p-value is greater than the significance level. D) Accept the null hypothesis since the p-value is less than the significance level.

Fill in the missing entries in the following partially completed one-way ANOVA table. \[\begin{array} { | l | l | l | l | l | } \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & & 28.9 & & \\ \hline \text { Error } & 30 & & 3.5 & \\ \hline \text { Total } & 33 & & & \\ \hline \end{array}\]

A) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-stati stic } \\ \hline \text { Treatment } & 63 & 28.9 & 0.46 & 520.20 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ B) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 2.75 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ C) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \mathrm{df} & \mathrm{SS} & \mathrm{MS}=\mathrm{SS} / \mathrm{df} & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 0.36 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ D) $\begin{array}{|l|r|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 2.75 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 29.02 & & \\ \hline \end{array}$ A) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-stati stic } \\ \hline \text { Treatment } & 63 & 28.9 & 0.46 & 520.20 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ B) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 2.75 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ C) $\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \mathrm{df} & \mathrm{SS} & \mathrm{MS}=\mathrm{SS} / \mathrm{df} & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 0.36 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 133.9 & & \\ \hline \end{array}$ D) $\begin{array}{|l|r|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 3 & 28.9 & 9.63 & 2.75 \\ \hline \text { Error } & 30 & 105.0 & 3.5 & \\ \hline \text { Total } & 33 & 29.02 & & \\ \hline \end{array}$

You roll a die 48 times with the following results. \[\begin{array} { c | l | l | r | r | r | r } \text { Number } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Frequency } & 2 & 4 & 12 & 13 & 14 & 3 \end{array}\] Use a significance level of 0.05 to test the claim that the die is fair.

The answer of You roll a die 48 times with...

Identify the p-value. \[\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\ \text { Error } & 8 & 50 & 6.25 & & \\ \text { Total } & 11 & 80 & & & \end{array}\]

A) 0.264 B) 1.6 C) 10.00 D) 6.25 A) 0.264 B) 1.6 C) 10.00 D) 6.25

Exam 11: Chi-Square and Analysis of Variance

A company manager wishes to test a union leader's claim that absences occur on the different week days with the same frequencies. Test this claim at the 0.05 level of significance if the following sample data have been compiled. Day Mon Tue Wed Thur Fri absences 37 15 12 23 43

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Question 1

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$\mathrm { H } _ { 0 }$ : The proportions of absences are all the same.
$\mathrm { H } _ { 1 }$ : The proportions of absences are not all the same.
Test statistic: $\chi ^ { 2 } = 28.308$ . Critical value: $\chi ^ { 2 } = 9.488$ . Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that absences occur on the different week days with the same frequency.

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -Random samples of four different models of cars were selected and the gas mileage of each car was measured. The results are shown below. Model A Model B Model C Model D 23 28 30 25 25 26 28 26 24 29 32 25 26 30 27 28 Test the claim that the four different models have the same population mean. Use a significance level of 0.05.

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Question 2

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Test statistic: $\mathrm { F } = 6.435$ . Critical value: $\mathrm { F } = 3.4903$ . P-value: $\mathrm { p } = 0.00762$ .
Reject the claim of equal means. The different models do not appear to have the same mean.

Use a $\chi ^ { 2 }$ 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 3

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$\mathrm { H } _ { 0 }$ : Employment status and response are independent.
$\mathrm { H } _ { 1 }$ : Employment status and response are dependent.
Test statistic: $\chi ^ { 2 } = 5.942$ . Critical value: $\chi ^ { 2 } = 4.605$ .
Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that response and employment status are independent.

Draw an example of an F distribution and list the characteristics of the F distribution.

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Question 4

At a high school debate tournament, half of the teams were asked to wear suits and ties and the rest were asked to wear jeans and t-shirts. The results are given in the table below. Test the hypothesis at the 0.05 level that the proportion of wins is the same for teams wearing suits as for teams wearing jeans. Win Loss Suit 22 28 T-shirt 28 22

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Question 5

At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results have been obtained. Brand A Brand B Brand C Brand D 17 18 21 22 20 18 24 25 21 23 25 27 22 25 26 29 21 26 29 35 29 36 37

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Question 6

Use a $\chi ^ { 2 }$ 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 gender and the sample results are given below. At the 0.05 significance level, test the claim that response and gender are independent. Yes No Undecided Male 25 50 15 Female 20 30 10

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Question 7

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -Given the sample data below, test the claim that the populations have the same mean. Use a significance level of 0.05. Brand A Brand B Brand C Brand D =16 =16 =16 =16 =2.09 =3.48 =1.86 =2.84 =0.37 =0.61 =0.45 =0.53

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Question 8

An observed frequency distribution is as follows: Number of successes 0 1 2 Frequency 47 98 55 i) Assuming a binomial distribution with $\mathrm { n } = 2$ and $\mathrm { p } = 1 / 2$ , use the binomial formula to find the probability corresponding to each category of the table. ii) Using the probabilities found in part (i), find the expected frequency for each category. iii) Use a $0.05$ level of significance to test the claim that the observed frequencies fit a binomial distribution for which $\mathrm { n } = 2$ and $\mathrm { p } = 1 / 2$ .

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Question 9

What can you conclude about the equality of the population means? Source DF SS MS F p Factor 3 13.500 4.500 5.17 0.011 Error 16 13.925 0.870 Total 19 27.425

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Question 10

An observed frequency distribution of exam scores is as follows: Exam Score under 60 60-69 70-79 80-89 90-100 Frequency 30 30 140 60 40 i) Assuming a normal distribution with $\mu = 75$ and $\sigma = 15$ , find the probability of a randomly selected subject belonging to each class. (Use boundaries of $59.5,69.5,79.5,89.5$ , 100.) ii) Using the probabilities found in part (i), find the expected frequency for each category. iii) Use a $0.05$ significance level to test the claim that the exam scores were randomly selected from a normally distributed population with $\mu = 75$ and $\sigma = 15$ .

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Question 11

Describe the null and alternative hypotheses for one-way ANOVA. Give an example.

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Question 12

Fill in the missing entries in the following partially completed one-way ANOVA table. Source df SS MS=SS/df F-statistic Treatment 28.9 Error 30 3.5 Total 33

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Question 13

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, 9, 0, 70, 485, 189, 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 14

Use the data given below to verify that the t test for independent samples and the ANOVA method are equivalent. 85 74 81 72 73 65 91 83 64 59 i) Use a t test with a 0.05 significance level to test the claim that the two samples come from populations with the same means. ii) Use the ANOVA method with a 0.05 significance level to test the same claim. iii) Verify that the squares of the t test statistic and the critical value are equal to the F test statistic and critical value.

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Question 15

You roll a die 48 times with the following results. Number 1 2 3 4 5 6 Frequency 2 4 12 13 14 3 Use a significance level of 0.05 to test the claim that the die is fair.

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Question 16

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -At the 0.025 significance level, test the claim that the three brands have the same mean if the following sample results have been obtained. Brand A Brand B Brand C 32 27 22 34 24 25 37 33 32 33 30 22 36 21 39

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Question 17

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, 12, 0, 73, 482, 186, 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

In the chi-square test of independence, the formula used is $\chi ^ { 2 } = \frac { \Sigma ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } }$ . Discuss the meaning of $\mathrm { O }$ and $\mathrm { E }$ and explain the circumstances under which the $\chi ^ { 2 }$ values will be smaller or larger. What is the relationship between a significant $\chi ^ { 2 }$ value and the values of O and E?

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Question 19

Identify the p-value. Source DF SS MS F p Factor 3 30 10.00 1.6 0.264 Error 8 50 6.25 Total 11 80

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Question 20

A company manager wishes to test a union leader's claim that absences occur on the different week days with the same frequencies. Test this claim at the 0.05 level of significance if the following sample data have been compiled. Day Mon Tue Wed Thur Fri absences 37 15 12 23 43

Draw an example of an F distribution and list the characteristics of the F distribution.

At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results have been obtained. Brand A Brand B Brand C Brand D 17 18 21 22 20 18 24 25 21 23 25 27 22 25 26 29 21 26 29 35 29 36 37

What can you conclude about the equality of the population means? Source DF SS MS F p Factor 3 13.500 4.500 5.17 0.011 Error 16 13.925 0.870 Total 19 27.425

Describe the null and alternative hypotheses for one-way ANOVA. Give an example.

Fill in the missing entries in the following partially completed one-way ANOVA table. Source df SS MS=SS/df F-statistic Treatment 28.9 Error 30 3.5 Total 33

You roll a die 48 times with the following results. Number 1 2 3 4 5 6 Frequency 2 4 12 13 14 3 Use a significance level of 0.05 to test the claim that the die is fair.

Identify the p-value. Source DF SS MS F p Factor 3 30 10.00 1.6 0.264 Error 8 50 6.25 Total 11 80

Introduction to Statistics

Summarizing and Graphing Data

Statistics for Describing, Exploring, and Comparing Data

Probability

Discrete Probability Distributions

Normal Probability Distributions

Estimates and Sample Size

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

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