The F-distribution is the sampling distribution of the ratio of: $\begin{array}{|l|l|} \hline \text { A. } & \text { two normal population variances. } \\ \hline \text { B. } & \text { two normal population means. } \\ \hline \text { C. } & \begin{array}{l} \text { two sample variances, provided that the samples are independently drawn from two } \\ \text { normal populations. } \end{array} \\ \hline \text { D. } & \text { two sample variances, provided that the sample sizes are large. } \\ \hline \end{array}$

The answer of The F-distribution is the sampling distribution of...

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is: $\begin{array}{|l|l|} \hline \text { A. } & \text { a normal distribution. } \\ \hline \text { B. } & \text { Student } t \text {-distribution. } \\ \hline \text { C. } & \text { an } F \text {-distribution. } \\ \hline \text { D. } & \text { a chi-squared distribution. } \\ \hline \end{array}$

The answer of Two independent samples are drawn from two...

The sampling distribution of the ratio of two sample variances $S _ { 1 } ^ { 2 }$ / $s _ { 2 } ^ { 2 }$ is said to be F-distributed provided that: $\begin{array}{|l|l|} \hline A.&\text { the samples are independent.}\\ \hline B.&\text {the populations are normal with equal variances.}\\ \hline C.&\text {the samples are dependent and their sizes are large. }\\ \hline D.&\text { the samples are independently drawn from two normal populations.}\\ \hline \end{array}$

The answer of The sampling distribution of the ratio of...

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses? $\begin{array}{|l|l|} \hline A.&\mathrm{Ho}: \sigma^{2}=10\\&H_{A}: \sigma^{2}>10\\ \hline B.&\mathrm{H}_{o} \mathrm{~s}^{2}=10\\&\mathrm{H}_{\mathrm{A}}: \mathrm{s}^{2}>10 \\ \hline C.&\mathrm{H}_{O} \mathrm{~s}^{2}=8\\&\mathrm{H}_{A}: s^{2}>8\\ \hline D.&\text { Ho: } \sigma^{2}=8\\&\mathrm{H}_{\mathrm{A}}: \sigma^{2}>8\\ \hline \end{array}$

The answer of A study wants to investigate whether the...

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is: $\begin{array}{|l|l|} \hline \text { A. } & \text { normally distributed. } \\ \hline \text { B. } & \text { Student } t \text { distributed. } \\ \hline \text { C. } & \text { chi-squared distributed. } \\ \hline \text { D. } & \text { F distributed. } \\ \hline \end{array}$

The answer of The ratio of two independent chi-squared variables,...

How many degrees of freedom are used for an F statistic? $\begin{array}{|l|l|} \hline \mathrm{A} & \left(\mathrm{n}_{1}-1\right) \times\left(\mathrm{n}_{2}-1\right) \\ \hline \mathrm{B} & \mathrm{n}-5 \\ \hline \mathrm{C} & {\mathrm{n}-2}\\ \hline D.&\text { None of these choices are correct. }\\ \hline \end{array}$

The answer of How many degrees of freedom are used...

Exam 15: Inference About Population Variances

To find the value in a chi-squared distribution with 10 degrees of freedom such that the area to its left is 0.01, we find the point in the same distribution such that the area to its left is 0.99.

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Question 21

The F-distribution is the sampling distribution of the ratio of: A. two normal population variances. B. two normal population means. C. two sample variances, provided that the samples are independently drawn from two normal populations. D. two sample variances, provided that the sample sizes are large.

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Question 22

A statistician wants to test for the equality of means in two independent samples drawn from normal populations. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The data are as follows: Sample 1: 7 9 6 15 7 10 8 12 Sample 2: 2 25 9 15 10 18 5 22 27 3 Given the data above, can the statistician conclude at the 5% significance level that the required condition is not satisfied?

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Question 23

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.05 to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ . The two sample variances are $s _ { 1 } ^ { 2 } = 500 \text { and } s _ { 2 } ^ { 2 } = 900$ , and the sample sizes are $n _ { 1 } = 21 \text { and } n _ { 2 } = 31$ . The rejection region is F > 2.20 or F < 0.4255.

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Question 24

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is: A. a normal distribution. B. Student t -distribution. C. an F -distribution. D. a chi-squared distribution.

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Question 25

The test statistic employed to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ is $F = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ , which is F-distributed with $v _ { 1 } = n _ { 1 } - 1 \text { and } v _ { 2 } = n _ { 2 } - 1$ degrees of freedom, provided that the two populations are F-distributed.

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Question 26

The sampling distribution of the ratio of two sample variances $S _ { 1 } ^ { 2 }$ / $s _ { 2 } ^ { 2 }$ is said to be F-distributed provided that: A. the samples are independent. B. the populations are normal with equal variances. C. the samples are dependent and their sizes are large. D. the samples are independently drawn from two normal populations.

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Question 27

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance? A. $\mathrm { n } - 2$ B. $\mathrm { n } - \mathrm { k }$ C. $n - 1$ D. $\mathrm { n }$

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Question 28

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses? A. :=10 :>10 B. =10 :>10 C. =8 :>8 D. Ho: =8 :>8

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Question 29

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is: A. normally distributed. B. Student t distributed. C. chi-squared distributed. D. F distributed.

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Question 30

Which of the following statements is correct regarding the percentile points of the chi-squared distribution? $\begin{array}{|l|l|}\hline A.&\text { \chi^{2} _{0.99,12}=26.2170 }\\\hline B.&\text { $ \chi_{0.95,12}^{2}=0.102587 $.}\\\hline C.&\text {$ x_{0.95,12}^{2}=28.2995 $. }\\\hline D.&\text {$ \chi^{2} _{0.99,12}=3.57056 $. }\\\hline \end{array}$

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Question 31

How many degrees of freedom are used for an F statistic? -1 \times -1 -5 -2 D. None of these choices are correct.

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Question 32

The value in a chi-squared distribution with 4 degrees of freedom such that the area to its right is 0.99 is 0.29711.

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Question 33

The value in an F-distribution with $v _ { 1 } = 4$ and $v _ { 2 } = 8$ degrees of freedom such that the area to its left is 0.975 is 5.05.

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Question 34

The value in an F-distribution with $v _ { 1 } = 6$ and $v _ { 2 } = 9$ degrees of freedom such that the area to its right is 0.05 is 3.37.

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Question 35

An investor is considering two types of investment. She is quite satisfied that the expected return on investment 1 is higher than the expected return on investment 2. However, she is quite concerned that the risk associated with investment 1 is higher than that of investment 2. To help make her decision, she randomly selects seven monthly returns on investment 1 and 10 monthly returns on investment 2. She finds that the sample variances of investments 1 and 2 are 225 and 118, respectively. Can she infer at the 5% significance level that the population variance of investment 1 exceeds that of investment 2?

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Question 36

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 10, $s _ { 1 } ^ { 2 } =$ 40. $n _ { 2 } =$ 15, $S _ { 2 } ^ { 2 } =$ 20. Is there enough evidence at the 5% significance level to infer that the variance of population 1 is larger than the variance of population 2?

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Question 37

A statistician wants to test for the equality of means in two independent samples drawn from normal populations. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The data are as follows: Sample 1: 7 9 6 15 7 10 8 12 Sample 2: 2 25 9 15 10 18 5 22 27 3 Briefly describe what the interval estimate in the previous question tells you.

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Question 38

The value in a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 15.5073.

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Question 39

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of: A. 0.90 B. 0.05 C. 0.20 D. 0.10

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Question 40

To find the value in a chi-squared distribution with 10 degrees of freedom such that the area to its left is 0.01, we find the point in the same distribution such that the area to its left is 0.99.

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is: A. a normal distribution. B. Student t -distribution. C. an F -distribution. D. a chi-squared distribution.

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance? A. $\mathrm { n } - 2$ B. $\mathrm { n } - \mathrm { k }$ C. $n - 1$ D. $\mathrm { n }$

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses? A. :=10 :>10 B. =10 :>10 C. =8 :>8 D. Ho: =8 :>8

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is: A. normally distributed. B. Student t distributed. C. chi-squared distributed. D. F distributed.

How many degrees of freedom are used for an F statistic? -1 \times -1 -5 -2 D. None of these choices are correct.

The value in a chi-squared distribution with 4 degrees of freedom such that the area to its right is 0.99 is 0.29711.

The value in an F-distribution with $v _ { 1 } = 4$ and $v _ { 2 } = 8$ degrees of freedom such that the area to its left is 0.975 is 5.05.

The value in an F-distribution with $v _ { 1 } = 6$ and $v _ { 2 } = 9$ degrees of freedom such that the area to its right is 0.05 is 3.37.

The value in a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 15.5073.

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of: A. 0.90 B. 0.05 C. 0.20 D. 0.10

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Multiple Regression

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

Exam 15: Inference About Population Variances

To find the value in a chi-squared distribution with 10 degrees of freedom such that the area to its left is 0.01, we find the point in the same distribution such that the area to its left is 0.99.

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is: A. a normal distribution. B. Student t -distribution. C. an F -distribution. D. a chi-squared distribution.

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance? A. n−2\mathrm { n } - 2n−2 B. n−k\mathrm { n } - \mathrm { k }n−k C. n−1n - 1n−1 D. n\mathrm { n }n

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses? A. :=10 :>10 B. =10 :>10 C. =8 :>8 D. Ho: =8 :>8

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is: A. normally distributed. B. Student t distributed. C. chi-squared distributed. D. F distributed.

How many degrees of freedom are used for an F statistic? -1 \times -1 -5 -2 D. None of these choices are correct.

The value in a chi-squared distribution with 4 degrees of freedom such that the area to its right is 0.99 is 0.29711.

The value in an F-distribution with v1=4v _ { 1 } = 4v1​=4 and v2=8v _ { 2 } = 8v2​=8 degrees of freedom such that the area to its left is 0.975 is 5.05.

The value in an F-distribution with v1=6v _ { 1 } = 6v1​=6 and v2=9v _ { 2 } = 9v2​=9 degrees of freedom such that the area to its right is 0.05 is 3.37.

The value in a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 15.5073.

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Multiple Regression

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance? A. $\mathrm { n } - 2$ B. $\mathrm { n } - \mathrm { k }$ C. $n - 1$ D. $\mathrm { n }$

The value in an F-distribution with $v _ { 1 } = 4$ and $v _ { 2 } = 8$ degrees of freedom such that the area to its left is 0.975 is 5.05.

The value in an F-distribution with $v _ { 1 } = 6$ and $v _ { 2 } = 9$ degrees of freedom such that the area to its right is 0.05 is 3.37.