Question

55. At the $5\%$ significance level, do we reject the null hypothesis?
Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up $35$ hills. The first cyclist has a variance of $23.8$ and the second cyclist has a variance of $32.1$. The cyclists want to see if their variances are the same or different. Assume that commute times are normally distributed.

Short answer

The $P$-value is $0.058$greater than the significance level of $0.05,$, so the null hypothesis for the test will be rejected. .Hence, there is sufficient proof that the variance of grades for the first cyclist is higher than that of the second cyclist.

Step-by-step solution

Given information

The significance level is $5\%$. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up $35$hills. The first cyclist has a variance of $23.8$and the second cyclist has a variance of $32.1$. The cyclists want to see if their variances are the same or different.

Explanation

In statistic, the significance level at $5\%$is tested by using $Tr-83$calculator. The procedure is given as below:

Click on STAT $\to$arrow over the TESTS $\to$arrow down to $2$-sample F-test $\to$ENTER $\to$arrow to the Stats and press ENTER. It will be,

Explanation

Then, enter the values of $S{x}_1=38.1,n_1=15,S{x}_2=22.5$, and $n_2=15$then press ENTER. It will be shown as:

Explanation

The hypothesis as $H_1:{\sigma }_1<{\sigma }_2$and press ENTER. Then arrow down to calculate, press ENTER. Then the obtained output is,