Answer to Question #285862 in Statistics and Probability for jew

Question

At 1% level of significance, can he conclude that the mean daily expenses of the sales staffs are greater than the audit staffs. State the null and alternative hypothesis

Answer 1

Since, the question is incomplete, let's assume the data:

Sales Staff

105

120

140

160

100

170

160

140

130

160

180

190

Audit Staff

130

50

90

80

70

125

180

80

110

To find: At 0.05 significance level, can he conclude that the mean daily expenses of sales staff are greater than the audit staff?

Solution:

Let $\mu_1$ be the sample mean for population 1 (sales staff) and $\mu2$ be the sample mean for population 2(audit staff) then, the hypothesis tested is,

$H_0:\mu_1=\mu_2$

$Against$

$H_1:\mu_1\gt\mu_2$

To perform this hypothesis test we shall use the $p-value$ technique the compare this value with the given $\alpha =0.05$ level of significance.

By applying the 2 sample t-test in $R$ we can easily perform the test using the commands below,

x=c(105,120,140,160,100,170,160,140,130,160,180,190)

y=c(130,50,90,80,70,125,180,80,110)

t. test(x, y)

The output for these commands are as shown below.

    Welch Two Sample t-test

data: x and y

t = 2.8812, df = 14.066, p-value = 0.01204

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

 11.40976 77.75691

sample estimates:

mean of x mean of y 

 146.2500 101.6667 

From this output the $p-value$ is divided by 2 since we are performing a one sided test thus, $p-value=0.01204/2=0.00602$ and reject the null hypothesis if $p-value\lt\alpha$

Since $p-value=0.00602\lt\alpha=0.05,$ we reject the null hypothesis and conclude that evidence exist to show that the mean daily expenses of sales staff are greater than the audit staff.