$20.$ Rabbitfoot Community Bank makes a large num$-$ ber of home equity loans each year. The vice pres$-$ ident of loan administration wishes to determine if their time for paperwork processing is lower than the average time of their top competitor. A sample of $30$ loans taken at Rabbitfoot Bank yielded a mean of $38\mathrm{.}10$ minutes and a standard deviation of $2\mathrm{.}58$ minutes $($see the data in the Ch$06$Data Excel workbook$).$ Data obtained from competitor of $36$ applications indicates that the average time for processing an application is $39\mathrm{.}48$ minutes, with a standard deviation of $2\mathrm{.}48$ minutes.$$a$.$ Verify the calculation of the mean, standard deviation, and variance for each set of data using the Descriptive Statistics tool.$$b$.$ Test the null hypothesis that his bank$$s proces$-$ sing time is greater than or equal to the compe$-$ titor$$s average, versus the alternative hypothesis that the bank$$s time is less than the competitor at the $5$ percent significance level.