Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low tech industry with CEOs in the consumer products industry. HYPOTHESIS: MEAN X = MEAN $Y$ $\begin{array}{l} \text { SAMPLES SELECTED FROM RETURN }\\ \begin{array}{lll} \text { industry } 1 & \text { (low tech) } & \text { (NUMBER }=15 \text { ) } \\ \text { industry } 3 & \text { (consumer products) } & \text { (NUMBER }=15) \end{array} \end{array}$ $X =$ industry1 $\mathrm { Y } =$ industry3 SAMPLE MEAN OF $X = 157.286$ SAMPLE VARIANCE OF $X = 1563.45$ SAMPLE SIZE OF $X = 14$ SAMPLE MEAN OF $\mathrm { Y } = 217.583$ SAMPLE VARIANCE OF $Y = 1601.54$ SAMPLE SIZE OF $\mathrm { Y } = 12$ MEAN X - MEAN Y $= - 60.2976$ $t = - 4.23468$ $\mathrm { P } - \mathrm { VALUE } = \quad 0.000290753$ $\mathrm { P } - \mathrm { VALUE } / 2 = 0.000145377$ SD. ERROR $= 14.239$ If we conclude that the mean return-to-pay ratios of the consumer products and low tech CEOs are Equal when, in fact, a difference really does exist between the means, we would be making a__________.

A) Type I error B) correct decision C) Type II error D) Type III error A) Type I error B) correct decision C) Type II error D) Type III error

We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table: \[\begin{array}{l} \begin{array} { c | c c | c } &\text { Price }\\ \text { Supermarket } & \text { Brand 1 } & \text { Brand 2 } & \text { Difference } \\ \hline 1 & \$ 2.25 & \$ 2.30 & \$ - 0.05 \\ 2 & 2.47 & 2.45 & 0.02 \\ 3 & 2.38 & 2.44 & - 0.06 \\ 4 & 2.27 & 2.29 & - 0.02 \\ 5 & 2.15 & 2.25 & - 0.10 \\ 6 & 2.25 & 2.25 & 0.00 \\ 7 & 2.36 & 2.42 & - 0.06 \\ 8 & 2.37 & 2.40 & - 0.03 \\ \hline & \mathrm { x } _ { 1 } = 2.3125 & \mathrm { x } _ { 2 } = 2.3500 & \mathrm {~d} = - 0.0375 \\ & \mathrm {~s} _ { 1 } = 0.1007 & \mathrm {~s} _ { 2 } = 0.0859 & \mathrm {~s} _ { \mathrm { d } } = 0.0381 \end{array} \end{array}\] Find a 98% confidence interval for the difference in mean price of brand 1 and brand 2.

A) $0.0375 \pm 0.0347$ B) $0.0375 \pm 0.1393$ C) $0.0375 \pm 0.0404$ D) $0.0375 \pm 0.0471$ A) $0.0375 \pm 0.0347$ B) $0.0375 \pm 0.1393$ C) $0.0375 \pm 0.0404$ D) $0.0375 \pm 0.0471$

Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low-tech industry with CEOs in the consumer products industry. $\text { HYPOTHESIS: MEAN X = MEAN Y }$ $\text {SAMPLES SELECTED FROM RETURN}$ $\begin{array}{lll} \text { industry } 1 & \text { (low tech) } & \text { (NUMBER = 15) } \\ \text { industry } 3 & \text { (consumer products) } & \text { (NUMBER = 15) } \end{array}$ $\begin{array}{rl} \text {X} =& \text {industry 1}\\ \text { Y} =&\text {ndustry3}\\ \\ \text { SAMPLE MEAN OF X }=&157.286 \\ \text { SAMPLE VARIANCE OF } X=&1563.45\\ \text { SAMPLE SIZE OF } X=&14 \\ \text { SAMPLE MEAN OF } \mathrm{Y}=&217.583\\ \text { SAMPLE VARIANCE OF} Y=&1601.54 \\ \text { SAMPLE SIZE OF}Y=&12 \\ \\ \text { MEAN X - MEAN Y }=&-60.2976 \\ \text {t}=&-4.23468 \\ \mathrm{P}-\mathrm{VALUE}=&0.000290753 \\ \text { P-VALUE } / 2=&0.000145377 \\ \mathrm{SD}, \mathrm{ERROR}=&14.239 \\ \end{array}$ Using the printout, which of the following assumptions is not necessary for the test to be valid?

A) The population variances are equal. B) The samples were randomly and independently selected. C) The population means are equal. D) Both populations have approximately normal distributions. A) The population variances are equal. B) The samples were randomly and independently selected. C) The population means are equal. D) Both populations have approximately normal distributions.

Exam 8: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low-tech industry and CEOs in the consumer products industry. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM RETURN industry $1 \quad$ (low tech) $\quad ( \mathrm { NUMBER } = 15 )$ industry $3 \quad$ (consumer products) $\quad ( \mathrm { NUMBER } = 15$ ) $X =$ industry1 $\mathrm { Y } =$ industry3 SAMPLE MEAN OF $X = 157.286$ SAMPLE VARIANCE OF $X = 1563.45$ SAMPLE SIZE OF X $= 14$ SAMPLE MEAN OF $Y = 217.583$ SAMPLE VARIANCE OF $Y = 1601.54$ SAMPLE SIZE OF $Y = 12$ MEAN X - MEAN Y $= - 60.2976$ $\mathrm { t } = - 4.23468$ $\mathrm { P } - \mathrm { VALUE } = 0.000290753$ P-VALUE $/ 2 = 0.000145377$ SD. ERROR $= 14.239$ Using the printout above, find the test statistic necessary for testing whether the mean return-to-pay ratio of low tech CEO's exceeds the return-to-pay ratio of consumer product CEOs.

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Question 41

The owners of an industrial plant want to determine which of two types of fuel (gas or electricity) will produce more useful energy at a lower cost. The cost is measured by plant investment per delivered quad (\$ invested /quadrillion BTUs). The smaller this number, the less the industrial plant pays for delivered energy. Random samples of 11 similar plants using electricity and 16 similar plants using gas were taken, and the plant investment/quad was calculated for each. In an analysis of the difference of means of the two samples, the owners were able to reject $H _ { 0 }$ in the test $H _ { 0 } : \left( \mu _ { E } - \mu _ { G } \right) = 0$ vs. $H _ { a } : \left( \mu _ { E } - \mu _ { G } \right) > 0$ . What is our best interpretation of the result?

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Question 42

The data for a random sample of six paired observations are shown below. Pair Observation 1 Observation 2 1 1 3 2 2 4 3 3 5 4 4 6 5 5 7 6 6 8 a. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1. Use the differences to calculate $s d ^ { 2 }$ . b. Calculate the standard deviations $s _ { 1 } ^ { 2 }$ and $s _ { 2 } ^ { 2 }$ of each column of observations. Then find pooled estimate of the variance $s p ^ { 2 }$ . c. Comparing $s ^ { 2 } d ^ { 2 }$ and $s p ^ { 2 }$ , explain the benefit of a paired difference experiment.

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Question 43

A marketing study was conducted to compare the variation in the age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. The sample data is shown here: Female: $\mathrm { n } = 31 , \quad$ sample mean $= 50.30 , \quad$ sample standard deviation $= 13.215$ Male: $\quad \mathrm { n } = 21 , \quad$ sample mean $= 39.80 , \quad$ sample standard deviation $= 10.040$ Identify the rejection region to that should be used to determine if the variation in the female ages exceeds the variation in the male ages when testing at $\alpha = 0.05$ .

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Question 44

An experiment has been conducted at a university to compare the mean number of study hours expended per week by student athletes with the mean number of hours expended by non athletes. A random sample of 55 athletes produced a mean equal to 20.6 hours studied per week and a standard deviation equal to 5.8 hours. A second random sample of 200 non athletes produced a mean equal to 23.5 hours per week and a standard deviation equal to 4.6 hours. How many students would need to be sampled in order to estimate the difference in means to within 1.5 hours with probability 95%?

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Question 45

Identify the rejection region that should be used to test $H _ { 0 } : \sigma _ { 1 } { } ^ { 2 } = \sigma _ { 2 } { } ^ { 2 }$ against $H _ { a } : \sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } { } ^ { 2 }$ for $v _ { 1 } = 5 , v _ { 2 } = 8$ , and $\alpha = .05$ .

(Multiple Choice)

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Question 46

Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low tech industry with CEOs in the consumer products industry. HYPOTHESIS: MEAN X = MEAN $Y$ SAMPLES SELECTED FROM RETURN industry 1 (low tech) (NUMBER =15 ) industry 3 (consumer products) (NUMBER =15) $X =$ industry1 $\mathrm { Y } =$ industry3 SAMPLE MEAN OF $X = 157.286$ SAMPLE VARIANCE OF $X = 1563.45$ SAMPLE SIZE OF $X = 14$ SAMPLE MEAN OF $\mathrm { Y } = 217.583$ SAMPLE VARIANCE OF $Y = 1601.54$ SAMPLE SIZE OF $\mathrm { Y } = 12$ MEAN X - MEAN Y $= - 60.2976$ $t = - 4.23468$ $\mathrm { P } - \mathrm { VALUE } = \quad 0.000290753$ $\mathrm { P } - \mathrm { VALUE } / 2 = 0.000145377$ SD. ERROR $= 14.239$ If we conclude that the mean return-to-pay ratios of the consumer products and low tech CEOs are Equal when, in fact, a difference really does exist between the means, we would be making a__________.

(Multiple Choice)

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Question 47

Given $v _ { 1 } = 15 \text { and } v _ { 2 } = 20 \text {, find } P ( F > 1.84 )$

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Question 48

When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Find the rejection region that would be used if it is desired to test to Determine if a difference exists between the proportion of the females and males who responds to the call to donate blood. $\text { Use } \alpha = 0.10$

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Question 49

When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Suppose 45 of the females and 60 of the males responded that they were able to give blood. Find the test statistic that would be used if it is desired to test to determine if a difference exists between the proportion of the females and males who responds to the call to donate blood.

(Multiple Choice)

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Question 50

In order for the results of a paired difference experiment to be unbiased, the experimental units in each pair must be chosen independently of one another.

(True/False)

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Question 51

Independent random samples selected from two normal populations produced the following sample means and standard deviations. Sample 1 Sample 2 =14 =11 =7.1 =8.4 =2.3 =2.9 Find and interpret the $95 \%$ confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ .

(Essay)

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Question 52

We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table: Price Supermarket Brand 1 Brand 2 Difference 1 \ 2.25 \ 2.30 \ -0.05 2 2.47 2.45 0.02 3 2.38 2.44 -0.06 4 2.27 2.29 -0.02 5 2.15 2.25 -0.10 6 2.25 2.25 0.00 7 2.36 2.42 -0.06 8 2.37 2.40 -0.03 =2.3125 =2.3500 =-0.0375 =0.1007 =0.0859 =0.0381 Find a 98% confidence interval for the difference in mean price of brand 1 and brand 2.

(Multiple Choice)

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Question 53

$\text { The sample mean difference } \bar { d } \text { is equal to the difference of the sample means } \bar { x } _ { 1 } - \bar { x } _ { 2 } \text {. }$

(True/False)

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Question 54

Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low-tech industry with CEOs in the consumer products industry. $\text { HYPOTHESIS: MEAN X = MEAN Y }$ $\text {SAMPLES SELECTED FROM RETURN}$ industry 1 (low tech) (NUMBER = 15) industry 3 (consumer products) (NUMBER = 15) X = industry 1 Y = ndustry3 SAMPLE MEAN OF X = 157.286 SAMPLE VARIANCE OF X= 1563.45 SAMPLE SIZE OF X= 14 SAMPLE MEAN OF = 217.583 SAMPLE VARIANCE OF Y= 1601.54 SAMPLE SIZE OF Y= 12 MEAN X - MEAN Y = -60.2976 t = -4.23468 -= 0.000290753 P-VALUE /2= 0.000145377 ,= 14.239 Using the printout, which of the following assumptions is not necessary for the test to be valid?

(Multiple Choice)

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Question 55

A paired difference experiment has 15 pairs of observations. What is the rejection region for testing $H _ { a } : \mu _ { d } > 0 \text { ? Use } \alpha = .05$ .

(Essay)

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Question 56

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } 2$ and $n _ { 1 } = n _ { 2 } = 20$ .

(Multiple Choice)

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Question 57

In an exit poll, 42 of 75 men sampled supported a ballot initiative to raise the local sales tax to build a new football stadium. In the same poll, 41 of 85 women sampled supported the initiative. Find and interpret the p-value for the test of hypothesis that the proportions of men and women who support the initiative are different.

(Essay)

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Question 58

Independent random samples, each containing 1,000 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 475 and 550 successes, respectively. Test $H _ { 0 } : \left( p _ { 1 } - p _ { 2 } \right) = 0$ against $H _ { \mathrm { a } } : \left( p _ { 1 } - p _ { 2 } \right) \neq 0$ . Use $\alpha = .01$ .

(Essay)

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Question 59

An inventor has developed a new spray coating that is designed to improve the wear of bicycle tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: Bicycle Coated Tire (C) Non-Coated Tire (N) 1 1.452 0.785 2 1.634 0.844 \downarrow \downarrow \downarrow 50 1.211 0.954 Coated Non-Coated Difference Mean 1.38 0.85 0.53 Std. Dev. 0.12 0.11 0.06 Sample Size 50 50 50 Use the summary data to construct a 90% confidence interval for the difference between the means.

(Multiple Choice)

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Question 60

Identify the rejection region that should be used to test $H _ { 0 } : \sigma _ { 1 } { } ^ { 2 } = \sigma _ { 2 } { } ^ { 2 }$ against $H _ { a } : \sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } { } ^ { 2 }$ for $v _ { 1 } = 5 , v _ { 2 } = 8$ , and $\alpha = .05$ .

Given $v _ { 1 } = 15 \text { and } v _ { 2 } = 20 \text {, find } P ( F > 1.84 )$

In order for the results of a paired difference experiment to be unbiased, the experimental units in each pair must be chosen independently of one another.

Independent random samples selected from two normal populations produced the following sample means and standard deviations. Sample 1 Sample 2 =14 =11 =7.1 =8.4 =2.3 =2.9 Find and interpret the $95 \%$ confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ .

$\text { The sample mean difference } \bar { d } \text { is equal to the difference of the sample means } \bar { x } _ { 1 } - \bar { x } _ { 2 } \text {. }$

A paired difference experiment has 15 pairs of observations. What is the rejection region for testing $H _ { a } : \mu _ { d } > 0 \text { ? Use } \alpha = .05$ .

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } 2$ and $n _ { 1 } = n _ { 2 } = 20$ .

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Simple Linear Regression

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Nonparametric Statistics Available on CD

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Exam 8: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Given v1=15 and v2=20, find P(F>1.84)v _ { 1 } = 15 \text { and } v _ { 2 } = 20 \text {, find } P ( F > 1.84 )v1​=15 and v2​=20, find P(F>1.84)

In order for the results of a paired difference experiment to be unbiased, the experimental units in each pair must be chosen independently of one another.

A paired difference experiment has 15 pairs of observations. What is the rejection region for testing Ha:μd>0 ? Use α=.05H _ { a } : \mu _ { d } > 0 \text { ? Use } \alpha = .05Ha​:μd​>0 ? Use α=.05 .

Calculate the degrees of freedom associated with a small-sample test of hypothesis for (μ1−μ2)\left( \mu _ { 1 } - \mu _ { 2 } \right)(μ1​−μ2​) , assuming σ12≠σ22\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } 2σ12​=σ2​2 and n1=n2=20n _ { 1 } = n _ { 2 } = 20n1​=n2​=20 .

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Simple Linear Regression

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Nonparametric Statistics Available on CD

Filters

Given $v _ { 1 } = 15 \text { and } v _ { 2 } = 20 \text {, find } P ( F > 1.84 )$

A paired difference experiment has 15 pairs of observations. What is the rejection region for testing $H _ { a } : \mu _ { d } > 0 \text { ? Use } \alpha = .05$ .

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } 2$ and $n _ { 1 } = n _ { 2 } = 20$ .