Question

F Test If using the sample data in Data Set 1 “Body Data” in Appendix B for a test of the claim that heights of men and heights of women have different variances, we find that s= 7.48296 cm for women and s= 7.10098 cm for men.

a. Find the values \(s_1^2\) and \(s_2^2\) express them with appropriate units of measure.

b. Identify the null and alternative hypotheses.

c. Find the value of the Ftest statistic and round it to four decimal places.

d. The P-value for this test is 0.5225. What do you conclude about the stated claim?

Short answer

a.The value of \(s_1^2\) is equal to 55.99469 cm squared, and the value of \(s_2^2\) is equal to 50.42392 cm squared.

b. Null Hypothesis: The variance of the heights of women is equal to the variance of the heights of men.

Alternative Hypothesis: The variance of the heights of women is not equal to the variance of the heights of men.

c. The F-statistic is equal to 1.1105.

d. There is not sufficient evidence to support the claim that heights of men and heights of women have different variances

Step-by-step solution

Given information

The sample standard deviation of the heights of men is equal to 7.10098 cm. The sample standard deviation of the heights of women is 7.48296 cm.

Find the values of the sample variances

In general, the larger of the two sample variances is denoted by \(s_1^2\) while, the smaller of the two sample variances is denoted by \(s_2^2\).

Here, the values of the sample variances are computed as shown:

\(\begin{array}{c}{s^2}_{women} = {\left( {7.48296} \right)^2}\\ = 55.99469\end{array}\)

\(\begin{array}{c}{s^2}_{men} = {\left( {7.10098} \right)^2}\\ = 50.42392\end{array}\)

It can be observed that the sample variance corresponding to the heights of women is greater than the sample variance corresponding to the heights of men.

Thus, \(s_1^2 = 55.99469\;{\rm{c}}{{\rm{m}}^{\rm{2}}}\) and \(s_2^2 = 50.42392\;{\rm{c}}{{\rm{m}}^{\rm{2}}}\).

State the hypotheses

b.

To test the significance of the claim that the variances of the heights of women and men are not equal, the hypotheses are formulated as follows:

Null Hypothesis: The variance of the heights of women is equal to the variance of the heights of men.

\({H_0}:{\sigma _1} = {\sigma _2}\)

Alternative Hypothesis: The variance of the heights of women is not equal to the variance of the heights of men.

\({H_1}:{\sigma _1} \ne {\sigma _2}\)

State the test statistic

c.

The value of the test statistic is computed as follows:

\(\begin{array}{c}F = \frac{{s_1^2}}{{s_2^2}}\\ = \frac{{55.99469}}{{50.42392}}\\ = 1.1105\end{array}\)

Thus, the F-statistic is equal to 1.1105.

Conclusion of the claim

d.

The level of significance assumed is equal to 0.05.

The p-value is equal to 0.5225, which is greater than 0.05. So, the null hypothesis is failed to reject.

Therefore, there is not sufficient evidence to support the claim that heights of men and heights of women have different variances