Chapter 11.5, Problem 21E

To determine

(a)

To State:

Null hypotheses and alternative hypotheses.

Answer to Problem 21E

Solution:

Null Hypotheses

${\text{H}}_0{\text{: σ}}_{\text{1}}^{\text{2}}\ge {\text{ σ}}_{\text{2}}^{\text{2}}$

Alternative Hypotheses

${\text{H}}_a{\text{ : σ}}_{\text{1}}^{\text{2}}{\text{ < σ}}_{\text{2}}^{\text{2}}$

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 ${\sigma }_1^2$ and population variances of the values of driving distances by the Generic golf balls be represented by ${\sigma }_2^2$. 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 ${\text{σ}}_{\text{1}}^{\text{2}}{\text{ < σ}}_{\text{2}}^{\text{2}}$. The mathematical opposite of this claim is ${\text{σ}}_{\text{1}}^{\text{2}}\ge {\text{ σ}}_{\text{2}}^{\text{2}}$. The null hypothesis and alternative hypothesis are stated as follows-

${\text{H}}_0{\text{: σ}}_{\text{1}}^{\text{2}}\ge {\text{ σ}}_{\text{2}}^{\text{2}}$

${\text{H}}_a{\text{ : σ}}_{\text{1}}^{\text{2}}{\text{ < σ}}_{\text{2}}^{\text{2}}$

To determine

(b)

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

Answer to Problem 21E

Solution:

The F- test statistic is appropriate and the level of significance for this test is $\alpha \text{ = 0}\text{.05}$.

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 $\alpha \text{ = 0}\text{.05}$.

Therefore, the F-test statistic is appropriate and the level of significance for this test is $\alpha \text{ = 0}\text{.05}$.

To determine

(c)

To Calculate:

The test statistic.

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,

$F=\frac{s_1^2}{s_2^2}$

Where $s_1^2$ and $s_2^2$ are the sample variances.

$n_1$ is the total number of data of population 1.

$n_2$ is the total number of data of population 2.

The degree of freedom for the numerator is $d{f}_1=n_1-1$

The degree of freedom for the denominator is $d{f}_2=n_2-1$

Calculation:

Given $n_1=10,s_1^2=201.65,n_2=10,s_2^2=364.57,\alpha =0.05$

The Null Hypothesis is,

${\text{H}}_0{\text{: σ}}_{\text{1}}^{\text{2}}\ge {\text{ σ}}_{\text{2}}^{\text{2}}$

Alternative Hypotheses –

${\text{H}}_a{\text{ : σ}}_{\text{1}}^{\text{2}}{\text{ < σ}}_{\text{2}}^{\text{2}}$

The test statistic value is given by,

$\text{F = }\frac{{\text{s}}_{\text{1}}^{\text{2}}}{{\text{s}}_{\text{2}}^{\text{2}}}$

$\begin{array}{c}\text{F = }\frac{\text{201}\text{.65}}{\text{364}\text{.57}}\\ \text{= 0}\text{.5531}\end{array}$

Therefore, the test statistic is 0.5531.

To determine

(d)

To Draw:

The conclusion and interpret the decision.

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,

$F\le F_{\left(1-\alpha \right)}$ for the left tailed test.

$F\ge F_{\alpha }$ for the right tailed test.

$F\le F_{\left(1-\frac{\alpha }{2}\right)}$ or $F\ge F_{\alpha /2}$ for the two tail test.

Calculation:

The level of significance $\text{α = 0}\text{.05}$

The degree of freedom for numerator is,

$d{f}_1=n_1-1$

Substitute 10 for $n_1$ in the above equation.

$\begin{array}{rll}d{f}_1& =& n_1-1\\ & =& 10-1\\ & =& 9\end{array}$

The degree of freedom for denominator is,

$d{f}_2=n_2-1$

Substitute 9 for $n_2$ in the above equation.

$\begin{array}{rll}d{f}_2& =& n_2-1\\ & =& 10-1\\ & =& 9\end{array}$

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

$\begin{array}{rll}{F}_{(1-\alpha )}& =& F_{(1-0.05)}\\ & =& F_{0.95}\\ & =& 0.3145\end{array}$

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.