The following data lists the ages of a random selection of actresses when they won an award in the category of Best Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts (a) and (b) below.Actress left parenthesis years right parenthesis25252827322525432836Actor left parenthesis years right parenthesis65353834293755424044a. Use the sample data with a 0.05 significance level to test the claim that for the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0 (indicating that the Best Actresses are generally younger than Best Actors).

The paired t - test checks the mean difference between matched (or) paired data. The t- test statistic formula of paired t test defined by t=xdsdn~(n-1)df Where x̄ d is the mean of difference b/w two paired groups Sd is the standard deviation of the difference n is the sample sizeState Claim : The population mean difference for ages of Best Actresses and Best Actors is less than zero(0) State hypotheses : Null hypothesis H0: μd = 0 ; Population mean difference for ages of Best Actresses and Best Actors is 0 Alternative hypothesis H1: μd < 0; Population mean difference for ages of Best Actresses and Best Actors is less than 0 Compute the t -test statistic : d = difference between two groups ages Actress (xi) Actor (yi) di = xi-yi 25 65 -40 25 35 -10 28 38 -10 27 34 -7 32 29 3 25 37 -12 25 55 -30 43 42 1 28 40 -12 36 44 -8 ∑di =-125 We have values, ∑di =-125 & n = 10 Find the value of x̄d xd=∑din =-12510 =-12.5 then, Find the value of Sd sd=∑di-xd2n-1 =1548.510-1 =13.117 Substitute the values x̄d = -12.5, Sd = 13.117 & n = 10 in the above defined t - formula t=-12.513.11710=-12.513.117/3.1623=-12.54.148=-3.014 Therefore, the value of t - test statistic is -3.014 Calculation of p value: if t test statistic is -3.014 with n-1= 9 degrees of freedom then p value is 0.0073 [ use t- distribution table values] Therefore, the p value for the test is 0.0073 Rejection rule : if p value < significance level then null hypothesis Ho should be rejected. If not then null hypothesis Ho should not be rejected. Given significance level is 0.05 Conclusion : Thus, the obtained p value (0.0073) < 0.05. So, null hypothesis Ho should be rejected at 0.05significance level. Hence, it is concluding that the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0.