Question 1

Which of the following statements are not true about the F distribution with parameters $v _ { 1 } \text { and } v _ { 2 } \text { ? }$

Accepted Answer

A)  The parameter $v _ { 1 }$
Is called the number of numerator degrees of freedom. 
B)  The parameter $v _ { 2 }$
Is called the number of denominator degrees of freedom. 
C)  A random variable that has an F distribution can assume a negative value; depends on the values of $v _ { 1 } \text { and } v _ { 2 }$ 
D)  All of the above statements are true. 
E)  None of the above statements are true. 
A)  The parameter $v _ { 1 }$
Is called the number of numerator degrees of freedom. 
B)  The parameter $v _ { 2 }$
Is called the number of denominator degrees of freedom. 
C)  A random variable that has an F distribution can assume a negative value; depends on the values of $v _ { 1 } \text { and } v _ { 2 }$ 
D)  All of the above statements are true. 
E)  None of the above statements are true.

Question 2

Give as much information as you can about the P-value of the F test in each of the following situations:
a. $v _ { 1 } = 5 , v _ { 2 } = 10 , \text { upper-tailed test, } f = 3.08$ 
b. $v _ { 1 } = 5 , v _ { 2 } = 10 , \text { upper-tailed test, } f = 2.15$ 
c. $v _ { 1 } = 5 , v _ { 2 } = 10 , \text { two-tailed test, } f = 10.48$ 
d. $v _ { 1 } = 5 , v _ { 2 } = 10 , \text { lower-tailed test, } f = .16$ 
e. $v _ { 1 } = 35 , v _ { 2 } = 20 , \text { upper-tailed test, } f = 3.24$

Accepted Answer

a. Since the given f value of 3.08 falls

Question 3

Let $\mu _ { 1 }$ denote true average tread life for a premium brand of radial tire and let $\mu _ { 2 }$ denote the true average tread life for an economy brand of the same size. Test $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = 5000$ versus $H _ { a } : \mu _ { 1 } - \mu _ { 2 } > 5000$ at level .01 using the following statistics: $m = 50 , \bar { x } = 43,000 , s _ { 1 } = 2200$ $n = 50 , \bar { y } = 37,000 , \text { and } s _ { 2 } = 1500$

Accepted Answer

The test statistic value is @#IMG-DLM& . @#IMG-DLM& will be

Question 4

Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$ with X and Y independent variables, and let $\hat { p } _ { 1 } = X / m \text { and } \hat { p } _ { 2 } = Y/ n$ Which of the following statements are not correct?

Accepted Answer

A)  $E \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } - p _ { 2 } \text {, so } \hat { p } _ { 1 } - \hat { p } _ { 2 }$
Is an unbiased estimator of $p _ { 1 } - p _ { 2 }$ 
B)  When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { \mathbf { 2 } }$
Individually has approximately normal distributions. 
C)  When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { 2 }$
Has approximately a normal distribution. 
D)  $V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n , \text { where } q _ { l } = 1 - p _ { l } \text { for } i = 1,2$ 
E)  All of the above statements are correct. 
A)  $E \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } - p _ { 2 } \text {, so } \hat { p } _ { 1 } - \hat { p } _ { 2 }$
Is an unbiased estimator of $p _ { 1 } - p _ { 2 }$ 
B)  When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { \mathbf { 2 } }$
Individually has approximately normal distributions. 
C)  When both m and n are large, the estimator $\hat { p } _ { 1 } - \hat { p } _ { 2 }$
Has approximately a normal distribution. 
D)  $V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n , \text { where } q _ { l } = 1 - p _ { l } \text { for } i = 1,2$ 
E)  All of the above statements are correct.

Question 5

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { \mathrm { m } }$ be a random sample from a normal population with mean $\mu _ { 1 } \text { and known variance } \sigma _ { 1 } ^ { 2 } \text {, }$ and let $Y _ { 1 } , Y _ { 2 } , \ldots \ldots , Y _ { n }$ be a random sample from a normal population with mean $\mu _ { 2 } \text { and variance } \sigma _ { 2 } ^ { 2 } \text {, }$ and that the X and Y samples are independent of one another. Which of the following statements are true?

Accepted Answer

A)  $\bar { X }$
Is normally distributed with expected value $\mu _ { 1 } \text { and variance }σ _ { 1 } ^ { 2 } / m$ 
B)  $\bar { Y }$
Is normally distributed with expected value $\mu_{2} \text { and variance } σ_{2}^{2} / n$ 
C)  $\bar { X } - \bar { Y }$
Is normally distributed with expected value $\mu _ { 1 } - \mu _ { 2 } \text { and variance } \left(σ _ { 1 } ^ { 2 } / m + \sigma _ { 2 } ^ { 2 } / n \right) \text {. }$ 
D)  $\bar { X } - \bar { Y }$
Is an unbiased estimator of $\mu _ { 1 } - \mu _ { 2 }$
) 
E)  All of the above statements are true. 
A)  $\bar { X }$
Is normally distributed with expected value $\mu _ { 1 } \text { and variance }σ _ { 1 } ^ { 2 } / m$ 
B)  $\bar { Y }$
Is normally distributed with expected value $\mu_{2} \text { and variance } σ_{2}^{2} / n$ 
C)  $\bar { X } - \bar { Y }$
Is normally distributed with expected value $\mu _ { 1 } - \mu _ { 2 } \text { and variance } \left(σ _ { 1 } ^ { 2 } / m + \sigma _ { 2 } ^ { 2 } / n \right) \text {. }$ 
D)  $\bar { X } - \bar { Y }$
Is an unbiased estimator of $\mu _ { 1 } - \mu _ { 2 }$
) 
E)  All of the above statements are true.

Question 6

A sample of 300 urban adult residents of in Michigan revealed 63 who favored increasing the highway speed limit from 55 to 70mph, whereas a sample of 180 rural residents yielded 72 who favored the increase. Does this data indicate that the sentiment for increasing the speed limit is different for the two groups of residents? Test $H _ { 0 } : p _ { 1 } = p _ { 2 } \text { versus } H _ { a } : p _ { 1 } \neq p _ { 2 }$ using $\alpha = .05$ , where $p _ { 1 }$ refers to the urban population.

Accepted Answer

@#IMG-DLM& will be rejected if @#IMG-DLM& @#IMG-DLM& , @#IMG-DLM& The

Question 7

Which of the following statements are true?

Accepted Answer

A)  Methods for comparing two population variances (or standard deviations) are occasionally needed, though such problems arise much less frequently than those involving means or proportions. 
B)  If $X _ { 1 } \text { and } X _ { 2 }$
Are independent chi-squared random variables with $v _ { 1 } \text { and } v _ { 2 }$
Degrees of freedom, respectively, divided by their respective degrees of freedom can be shown to have an F distribution. 
C)  The density curve of an F distribution is positively skewed (skewed to the right). 
D)  All of the above statements are true. 
E)  None of the above statements are true. 
A)  Methods for comparing two population variances (or standard deviations) are occasionally needed, though such problems arise much less frequently than those involving means or proportions. 
B)  If $X _ { 1 } \text { and } X _ { 2 }$
Are independent chi-squared random variables with $v _ { 1 } \text { and } v _ { 2 }$
Degrees of freedom, respectively, divided by their respective degrees of freedom can be shown to have an F distribution. 
C)  The density curve of an F distribution is positively skewed (skewed to the right). 
D)  All of the above statements are true. 
E)  None of the above statements are true.

Question 8

A 95% confidence interval for $\mu _ { D }$ the true mean difference in paired data, where $\bar { d } = 20 , s _ { D } = 12$ $\text { and } n = 15$ is determined by

Accepted Answer

A)  20 $\pm$
2)048 (0.80) 
B)  20 $\pm$
2)145 (3.098) 
C)  20 $\pm$
2)131 (0.894) 
D)  20 $\pm$
1)761 (1.118) 
E)  20 $\pm$
1)753(1.291) 
A)  20 $\pm$
2)048 (0.80) 
B)  20 $\pm$
2)145 (3.098) 
C)  20 $\pm$
2)131 (0.894) 
D)  20 $\pm$
1)761 (1.118) 
E)  20 $\pm$
1)753(1.291)

Question 9

Two independent samples of sizes m and n and variances $ { S } _ { 1 } ^ { 2 } \text { and } S _ { 2 } ^ { 2 }$ are selected at random from two normal distributions with variances $σ _ { 1 } ^ { 2 } \text { and } σ  _ { 2 } ^ { 2 } \text {. }$ In testing $H _ { o } : σ _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text { versus } H _ { o} : σ _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 } \text {, }$  where the test statistic value is $f = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 } ,$ the rejection region for a level .05 test is either $f \geq$$\underline{\quad\quad}$ or $f \leq$$\underline{\quad\quad}$

Accepted Answer

The answer of Two independent samples of sizes m and...

Question 10

Suppose $\mu _ { 1 } \text { and } \mu _ { 2 }$ are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. Use the two-sample $t$ test at significance level .01 to test $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = - 10 \text { versus } H _ { a } : \mu _ { 1 } - \mu _ { 2 } < - 10$ for the following statistics: $m = 6 , \bar { x } = 116 , s _ { 1 } = 5.0 , n = 6 , \bar { y } = 129 , \text { and } s _ { 2 } = 5.5$

Accepted Answer

For the given hypotheses, the

Question 11

Two different types of alloy, A and B, have been used to manufacture experimental specimens of a small tension link to be used in a certain engineering application. The ultimate strength (ksi) of each specimen was determined, and the results are summarized in the accompanying frequency distribution. $$\begin{array} { c c l } 
\hline & \text { A } & \text { B } \
\hline 26 - < 30 & 6 & 4 \
\hline 30 - < 34 & 12 & 9 \
\hline 34 - < 38 & 15 & 19 \
\hline 38 - < 42 & 7 & 10 \
\hline & m = 40 & n = 42 \
\hline
\end{array}$$ Compute a 95% CI for the difference between the true proportions of all specimens of alloys A and B that have an ultimate strength of at least 34 ksi.

Accepted Answer

Since @#IMG-DLM& the

Question 12

Tensile strength tests were carried out on two different grades of wire rod resulting in the accompanying data: $\begin{array}{llll}
\hline \text { Grade } & \text { Sample Size } &\text { Sample Mean } & \text { Sample St. Dev. }\ 
&&\left(\mathrm{kg} / \mathrm{mm}^{2}\right)\
\hline \text { AISI } & m= & \bar{x}= & s_{1}= \
1064 & 130 & 108 & 1.3 \
\text { AISI }& n= & \bar{y}= & s_{2}= \
1078   & 130 & 124 & 2.0 \
\hline 
\end{array}$
 
a. Does the data provide compelling evidence for concluding that "true" average strength for the 1078 grade exceeds that for the 1064 grade by more than 10 $\mathrm { kg } / \mathrm { mm } ^ { 2 }$ 
? Test the appropriate hypotheses using the $P$ 
-value approach. 
b. Estimate the difference between true average strengths for the two grades in a way that provides information about precision and reliability.

Accepted Answer

a. Parameter of interest: @#IMG-DLM& = the true di

Question 13

Suppose $\mu _ { 1 } \text { and } \mu _ { 2 }$ are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. The following statistics are given: m = 6, $\bar { x } = 116 , s _ { 1 } = 5.0 , n = 6 , \bar { y } = 129 \text {, and } s _ { 2 } = 5.5$ Calculate a 95% CI for the difference between true average stopping distance for cars equipped with system 1 and cars equipped with system 2. Does the interval suggest that precise information about the value of this difference is available?

Accepted Answer

We want a 95% confidence inter

Question 14

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { 0  } : p _ { 1 } - p _ { 2 } > 0 \text {, where } p _ { 1 } \text { and } p _ { 2 }$ denote the two population proportions, and both sample sizes are assumed to be large, the rejection region for approximate level .025 test is

Accepted Answer

A)  $z \leq - 1.96$ 
B)  $z \geq 1.96$ 
C)  either $z \leq - 1.96 \text { or } z \geq 1.96$ 
D)  $z \leq - 1.65$ 
E)  $z \geq 1.65$ 
A)  $z \leq - 1.96$ 
B)  $z \geq 1.96$ 
C)  either $z \leq - 1.96 \text { or } z \geq 1.96$ 
D)  $z \leq - 1.65$ 
E)  $z \geq 1.65$

Question 15

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { \text {at } } : p _ { 1 } - p _ { 2 } < 0 \text { where } p _ { 1 } \text { and } p _ { 2 }$ denote the two population properties, the P-value is found to be .0715. Then at .05 level, $H _ { o }$ should __________.

Accepted Answer

The answer of In testing \(H _ { 0 }...

Question 16

Let $X _ { 1 } , \ldots \ldots , X _ { m }$ be a random sample from a normal distribution with variance $\sigma _ { 1 } ^ { 2 } , \text { let } Y _ { 1 } , \ldots \ldots , Y _ { n }$ be another random sample (independent of the $\left. X _ { 2 } ^ { \prime } s \right)$ from a normal distribution with variance $\sigma _ { 2 } ^ { 2 } , \text { and }  \mathrm { let } S _ { 1 } ^ { 2 } \text { and } S _ { 2 } ^ { 2 }$ denote the two sample variances. Which of the following statements are not true?

Accepted Answer

A)  The random variable $F = \left( S _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } \right)/ \left( S _ { 2 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } \right)$
Has an F distribution with parameters $v _ { 1 } = m - 1 \text { and } v _ { 2 } = n - 1$ 
B)  The random variables $( m - 1 ) S _ { 1 } ^ { 2 } / σ _ { 1 } ^ { 2 } \text { and } ( n - 1 ) S _ { 2 } ^ { 2 } / σ _ { 2 } ^ { 2 }$ 
Each have a t distribution with m-1 and n-1 degrees of freedom, respectively. 
C)  The hypothesis $H _ { 0 } : σ _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ 
Is rejected if the ratio of the sample variances differs by too much from 1. 
D)  In testing $H _ { o } : σ  _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text { versus } H _ {0} : σ  _ { 1 } ^ { 2 } >σ  _ { 2 } ^ { 2 } \text {, }$
The rejection region for a level $\alpha \text { test is } f \geq F _ { \alpha , m - 1 n - 1 }$ 
E)  All of the above statements are true. 
A)  The random variable $F = \left( S _ { 1 } ^ { 2 } / \sigma _ { 1 } ^ { 2 } \right)/ \left( S _ { 2 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } \right)$
Has an F distribution with parameters $v _ { 1 } = m - 1 \text { and } v _ { 2 } = n - 1$ 
B)  The random variables $( m - 1 ) S _ { 1 } ^ { 2 } / σ _ { 1 } ^ { 2 } \text { and } ( n - 1 ) S _ { 2 } ^ { 2 } / σ _ { 2 } ^ { 2 }$ 
Each have a t distribution with m-1 and n-1 degrees of freedom, respectively. 
C)  The hypothesis $H _ { 0 } : σ _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ 
Is rejected if the ratio of the sample variances differs by too much from 1. 
D)  In testing $H _ { o } : σ  _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text { versus } H _ {0} : σ  _ { 1 } ^ { 2 } >σ  _ { 2 } ^ { 2 } \text {, }$
The rejection region for a level $\alpha \text { test is } f \geq F _ { \alpha , m - 1 n - 1 }$ 
E)  All of the above statements are true.

Question 17

Which of the following statements are not necessarily true?

Accepted Answer

A)  The density curve of an F distribution is not symmetric, so it would be necessary that both upper-and lower-tail critical values must be tabulated. 
B)  $F _ { 1 - \alpha , v _ { 1 } , v _ { 2 } } = 1 / F _ { \alpha , v _ { 1 } , v _ { 1 } }$ 
C)  There is an important connection between an F distribution and independent chi-squared random variables. 
D)  A random variable that has an F distribution cannot assume a negative value. 
E)  All of the above statements are true. 
A)  The density curve of an F distribution is not symmetric, so it would be necessary that both upper-and lower-tail critical values must be tabulated. 
B)  $F _ { 1 - \alpha , v _ { 1 } , v _ { 2 } } = 1 / F _ { \alpha , v _ { 1 } , v _ { 1 } }$ 
C)  There is an important connection between an F distribution and independent chi-squared random variables. 
D)  A random variable that has an F distribution cannot assume a negative value. 
E)  All of the above statements are true.

Question 18

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { \text {at } } : p _ { 1 } - p _ { 2 } > 0 \text { where } p _ { 1 } \text { and } p _ { 2 }$ denote the two population proportions, the following summary statistics are given: m = 400, x = 140, n = 500 and y = 160. Then the value of the test statistic is z = __________.

Accepted Answer

The answer of In testing \(H _ { 0 }...

Question 19

In an experiment designed to study the effects of illumination level on task performance, subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with black background and a higher level with a white background. Each data value is the time (sec) required to complete the task.
Subject $$\begin{array} { l | c c c c c c c c c } 
\hline & \mathbf { 1 } & \mathbf { 2 } & \mathbf { 3 } & \mathbf { 4 } & \mathbf { 5 } & \mathbf { 6 } & \mathbf { 7 } & \mathbf { 8 } & \mathbf { 9 } \
\hline \text { Black } & 25.01 & 41.05 & 27.47 & 25.74 & 24.96 & 28.84 & 25.85 & 20.89 & 32.05 \
\text { White } & 16.61 & 24.98 & 24.59 & 19.68 & 16.07 & 20.84 & 18.23 & 19.50 & 22.96 \
\hline
\end{array}$$ Compute in interval estimate for the difference between true average task time under the high illumination level and true average time under the low level.

Accepted Answer

The differences (white - black) are -7.6

Question 20

Which of the following statements are not true?

Accepted Answer

A)  Many statisticians recommend pooled t procedures over the two-sample t procedures. 
B)  The pooled t test is not a likelihood ratio test, whereas the two-sample t test can be derived from the likelihood ratio principle. 
C)  The significance level for the pooled t test is exact. 
D)  The significance level for the two-sample t test is only approximate. 
E)  All of the above statements are true 
A)  Many statisticians recommend pooled t procedures over the two-sample t procedures. 
B)  The pooled t test is not a likelihood ratio test, whereas the two-sample t test can be derived from the likelihood ratio principle. 
C)  The significance level for the pooled t test is exact. 
D)  The significance level for the two-sample t test is only approximate. 
E)  All of the above statements are true

Which of the following statements are not true about the F distribution with parameters $v _ { 1 } \text { and } v _ { 2 } \text { ? }$

Which of the following statements are true?

A 95% confidence interval for $\mu _ { D }$ the true mean difference in paired data, where $\bar { d } = 20 , s _ { D } = 12$ $\text { and } n = 15$ is determined by

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { 0 } : p _ { 1 } - p _ { 2 } > 0 \text {, where } p _ { 1 } \text { and } p _ { 2 }$ denote the two population proportions, and both sample sizes are assumed to be large, the rejection region for approximate level .025 test is

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { \text {at } } : p _ { 1 } - p _ { 2 } < 0 \text { where } p _ { 1 } \text { and } p _ { 2 }$ denote the two population properties, the P-value is found to be .0715. Then at .05 level, $H _ { o }$ should __________.

Which of the following statements are not necessarily true?

Which of the following statements are not true?

Overview and Descriptive Statistics

Probability

Discrete Random Variables and Probability Distributions

Continuous Random Variables and Probability Distributions

Joint Probability Distributions and Random Samples

Point Estimation

Statistical Intervals Based on a Single Sample

Tests of Hypotheses Based on a Single Sample

The Analysis of Variance

Multifactor Analysis of Variance

Simple Linear Regression and Correlation

Nonlinear and Multiple Regression

Goodness-Of-Fit Tests and Categorical Data Analysis

Distribution-Free Procedures

Quality Control Methods

Filters

Exam 9: Inferences Based on Two Samples

Which of the following statements are not true about the F distribution with parameters v1 and v2 ? v _ { 1 } \text { and } v _ { 2 } \text { ? }v1​ and v2​ ?

Which of the following statements are true?

A 95% confidence interval for μD\mu _ { D }μD​ the true mean difference in paired data, where dˉ=20,sD=12\bar { d } = 20 , s _ { D } = 12dˉ=20,sD​=12 and n=15\text { and } n = 15 and n=15 is determined by

Which of the following statements are not necessarily true?

Which of the following statements are not true?

Overview and Descriptive Statistics

Probability

Discrete Random Variables and Probability Distributions

Continuous Random Variables and Probability Distributions

Joint Probability Distributions and Random Samples

Point Estimation

Statistical Intervals Based on a Single Sample

Tests of Hypotheses Based on a Single Sample

The Analysis of Variance

Multifactor Analysis of Variance

Simple Linear Regression and Correlation

Nonlinear and Multiple Regression

Goodness-Of-Fit Tests and Categorical Data Analysis

Distribution-Free Procedures

Quality Control Methods

Filters

Which of the following statements are not true about the F distribution with parameters $v _ { 1 } \text { and } v _ { 2 } \text { ? }$

A 95% confidence interval for $\mu _ { D }$ the true mean difference in paired data, where $\bar { d } = 20 , s _ { D } = 12$ $\text { and } n = 15$ is determined by