Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
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Chapter 11.5, Problem 21E
To determine

(a)

To State:

Null hypotheses and alternative hypotheses.

Expert Solution
Check Mark

Answer to Problem 21E

Solution:

Null Hypotheses

H0: σ12  σ22

Alternative Hypotheses

Ha : σ12 < σ22

Explanation of Solution

Given:

A golf pro believes that the variances of his driving distances are different for different brands of golf balls. In particular, he believes that his driving distances, measured in yards, have a smaller variance when he uses Titleist golf balls than when he uses a generic store brand. He hits 10 Titleist golf balls and records a sample variance of 201.65. He hits 10 generic golf balls and records a sample variance of 364.57. Test the golf pro’s claim using a 0.05 level of significance.

Calculation:

Let population variances of the values of driving distances by the Titleist golf balls be represented by σ12 and population variances of the values of driving distances by the Generic golf balls be represented by σ22. A golf pro believes that his driving distances, measured in yards, have a smaller variance when he uses Titleist golf balls than when he uses a generic store brand that is σ12 < σ22. The mathematical opposite of this claim is σ12  σ22. The null hypothesis and alternative hypothesis are stated as follows-

H0: σ12  σ22

Ha : σ12 < σ22

To determine

(b)

The type of distribution to use for the test statistics and state the level of significance.

Expert Solution
Check Mark

Answer to Problem 21E

Solution:

The F- test statistic is appropriate and the level of significance for this test is α = 0.05.

Explanation of Solution

Given:

A golf pro believes that the variances of his driving distances are different for different brands of golf balls. In particular, he believes that his driving distances, measured in yards, have a smaller variance when he uses Titleist golf balls than when he uses a generic store brand. He hits 10 Titleist golf balls and records a sample variance of 201.65. He hits 10 generic golf balls and records a sample variance of 364.57. Test the golf pro’s claim using a 0.05 level of significance.

Calculation:

For comparing the variances of two normally distributed populations using independent, simple random samples, F- test statistic is used. The level of significance for this test is α = 0.05.

Therefore, the F-test statistic is appropriate and the level of significance for this test is α = 0.05.

To determine

(c)

To Calculate:

The test statistic.

Expert Solution
Check Mark

Answer to Problem 21E

Solution:

The test statistic is 0.5531.

Explanation of Solution

Given:

A golf pro believes that the variances of his driving distances are different for different brands of golf balls. In particular, he believes that his driving distances, measured in yards, have a smaller variance when he uses Titleist golf balls than when he uses a generic store brand. He hits 10 Titleist golf balls and records a sample variance of 201.65. He hits 10 generic golf balls and records a sample variance of 364.57. Test the golf pro’s claim using a 0.05 level of significance.

Formula used:

When the samples are given to be independent, the given population distribution are approximately normal, then the test statistics for the hypothesis test for two population variances is given by,

F=s12s22

Where s12 and s22 are the sample variances.

n1 is the total number of data of population 1.

n2 is the total number of data of population 2.

The degree of freedom for the numerator is df1=n11

The degree of freedom for the denominator is df2=n21

Calculation:

Given n1=10, s12= 201.65, n2=10, s22 = 364.57, α = 0.05

The Null Hypothesis is,

H0: σ12  σ22

Alternative Hypotheses –

Ha : σ12 < σ22

The test statistic value is given by,

F = s12s22

F = 201.65364.57= 0.5531

Therefore, the test statistic is 0.5531.

To determine

(d)

To Draw:

The conclusion and interpret the decision.

Expert Solution
Check Mark

Answer to Problem 21E

Solution:

The null hypothesis is accepted and it is concluded that there is a no sufficient evidence at the 0.05 level of significance to support the claim that variances of driving distances are different for different brands of golf balls.

Explanation of Solution

Given:

A golf pro believes that the variances of his driving distances are different for different brands of golf balls. In particular, he believes that his driving distances, measured in yards, have a smaller variance when he uses Titleist golf balls than when he uses a generic store brand. He hits 10 Titleist golf balls and records a sample variance of 201.65. He hits 10 generic golf balls and records a sample variance of 364.57. Test the golf pro’s claim using a 0.05 level of significance.

Formula used:

The null hypothesis is rejected if,

FF(1α) for the left tailed test.

FFα for the right tailed test.

FF(1α2) or FFα/2 for the two tail test.

Calculation:

The level of significance α = 0.05

The degree of freedom for numerator is,

df1=n11

Substitute 10 for n1 in the above equation.

df1= n1-1= 10-1= 9

The degree of freedom for denominator is,

df2=n21

Substitute 9 for n2 in the above equation.

df2= n2-1= 10-1= 9

From the F-table, the critical values for the required degrees is,

F(1- α)= F(1- 0.05)= F0.95= 0.3145

By comparing the test statistic value and the critical value, the F value is greater than the critical value, so by the left tailed test the null hypothesis is accepted.

Conclusion:

Thus, there is no sufficient evidence at 0.05 level of significance to support the claim that the variances of driving distances are different for different brands of golf balls.

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Chapter 11 Solutions

Beginning Statistics, 2nd Edition

Ch. 11.1 - Prob. 11ECh. 11.1 - Prob. 12ECh. 11.1 - Prob. 13ECh. 11.1 - Prob. 14ECh. 11.1 - Prob. 15ECh. 11.1 - Prob. 16ECh. 11.1 - Prob. 17ECh. 11.1 - Prob. 18ECh. 11.1 - Prob. 19ECh. 11.1 - Prob. 20ECh. 11.1 - Prob. 21ECh. 11.2 - Prob. 1ECh. 11.2 - Prob. 2ECh. 11.2 - Prob. 3ECh. 11.2 - Prob. 4ECh. 11.2 - Prob. 5ECh. 11.2 - Prob. 6ECh. 11.2 - Prob. 7ECh. 11.2 - Prob. 8ECh. 11.2 - Prob. 9ECh. 11.2 - Prob. 10ECh. 11.2 - Prob. 11ECh. 11.2 - Prob. 12ECh. 11.2 - Prob. 13ECh. 11.2 - Prob. 14ECh. 11.2 - Prob. 15ECh. 11.2 - Prob. 16ECh. 11.2 - Prob. 17ECh. 11.3 - Prob. 1ECh. 11.3 - Prob. 2ECh. 11.3 - Prob. 3ECh. 11.3 - Prob. 4ECh. 11.3 - Prob. 5ECh. 11.3 - Prob. 6ECh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.4 - Prob. 1ECh. 11.4 - Prob. 2ECh. 11.4 - Prob. 3ECh. 11.4 - Prob. 4ECh. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 10ECh. 11.4 - Prob. 11ECh. 11.5 - Prob. 1ECh. 11.5 - Prob. 2ECh. 11.5 - Prob. 3ECh. 11.5 - Prob. 4ECh. 11.5 - Prob. 5ECh. 11.5 - Prob. 6ECh. 11.5 - Prob. 7ECh. 11.5 - Prob. 8ECh. 11.5 - Prob. 9ECh. 11.5 - Prob. 10ECh. 11.5 - Prob. 11ECh. 11.5 - Prob. 12ECh. 11.5 - Prob. 13ECh. 11.5 - Prob. 14ECh. 11.5 - Prob. 15ECh. 11.5 - Prob. 16ECh. 11.5 - Prob. 17ECh. 11.5 - Prob. 18ECh. 11.5 - Prob. 19ECh. 11.5 - Prob. 20ECh. 11.5 - Prob. 21ECh. 11.5 - Prob. 22ECh. 11.5 - Prob. 23ECh. 11.5 - Prob. 24ECh. 11.5 - Prob. 25ECh. 11.5 - Prob. 26ECh. 11.6 - Prob. 1ECh. 11.6 - Prob. 2ECh. 11.6 - Prob. 3ECh. 11.6 - Prob. 4ECh. 11.6 - Prob. 5ECh. 11.6 - Prob. 6ECh. 11.6 - Prob. 7ECh. 11.6 - Prob. 8ECh. 11.6 - Prob. 9ECh. 11.6 - Prob. 10ECh. 11.6 - Prob. 11ECh. 11.6 - Prob. 12ECh. 11.6 - Prob. 13ECh. 11.6 - Prob. 14ECh. 11.6 - Prob. 15ECh. 11.6 - Prob. 16ECh. 11.CR - Prob. 1CRCh. 11.CR - Prob. 2CRCh. 11.CR - Prob. 3CRCh. 11.CR - Prob. 4CRCh. 11.CR - Prob. 5CRCh. 11.CR - Prob. 6CRCh. 11.CR - Prob. 7CRCh. 11.CR - Prob. 8CRCh. 11.CR - Prob. 9CRCh. 11.CR - Prob. 10CR
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