Question

Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for $20$commutes. The first worker’s times have a variance of $12.1$. These coworkers' times have a variance of $16.9$. The first worker thinks that he is more consistent with his commute times. Test the claim at the $10$% level. Assume that commute times are normally distributed. What is the p-value?

Short answer

The p-value is$0.47$

Step-by-step solution

Given Information

The following information is given in the problem, they each record their times for $20$commutes. The first worker’s times have a variance of $12.1$. These coworkers' times have a variance of $16.9$. Test the claim at the $10$% level. What is the p-value?

Explanation

Let 1 and 2 be the subscripts that indicate the first and second worker

$n_1=n_2=20$

A variance of the first worker's times is

$s_1^2=12.1$

A variance of the second worker's times is

$s_2^2=16.9$

$H_0:{\sigma }_1^2={\sigma }_2^2$and $H_a:{\sigma }_1^2<{\sigma }_2^2$

By the null hypothesis$\left({\sigma }_1^2={\sigma }_2^2\right)$, the$F$statistic is;

$F=\frac{\left[\frac{{\left(s_1\right)}^2}{{\left({\sigma }_1\right)}^2}\right]}{\left[\frac{{\left(s_1\right)}^2}{{\left({\sigma }_2\right)}^2}\right]}=\frac{{\left(s_1\right)}^2}{{\left(s_2\right)}^2}=\frac{12.1}{16.9}=0.716$

Distribution for the test: $F_{19,19}$where $n_1-1=19$and $n_2-1=19$

Probability statement: $p-$value $=P(F<0.716)=0.47$