Question

A survey of 410 workers showed that 172 said it was unethical to monitor employee e-mail. When 135 senior-level bosses were surveyed, 37 said it was unethical to monitor employee e-mail. At the 1% significance level, do the data provide sufficient evidence to conclude that the proportion of workers that say monitoring employee e-mail is unethical is greater than the proportion of bosses.

A. What is the Parameter of interest?
B. What is the underlying Distribution?

Answer 1

Explanation:

Analysis of the Data Given:

For sample 1 , the sample size
`N_1 = 410`

number of favorable cases
`X_1 = 172`

thus ; the sample proportion is
`\bar{p_1} = (X_1)/(N_1)`

=
`(172)/(410)`

= 0.4195

For sample 2, the sample size
`N_2 = 135`

number of favorable cases
`X_2 = 37`

Then the sample proportion is
`\bar{p_2} = (X_2)/(N_2)`

=
`(37)/(135)`

= 0.2741

The value of the pooled proportion is computed as
`\bar p = (X_1+X_2)/(N_1+N_2)`

=
`(172+37)/(410+135)`

= 0.3835

We are also given that the significance level is
`\alpha =0.01`

Null Hypothesis :
`H_0: p_1 =p_2`: the proportion of workers that say monitoring employee e-mail is unethical is not greater than the proportion of bosses.

Alternative hypothesis :
`H_1 : p_1 > p_2`: the proportion of workers that say monitoring employee e-mail is unethical is greater than the proportion of bosses.

`H_0: p_1 =p_2`:

`H_1 : p_1 > p_2`:

The above corresponds to the right-tailed test , for which a z-test for the two population proportions needs to be conducted .

Rejection Region:

From the given information ; the significance level is
`\alpha =0.01`

Then; the critical value for a right tailed test is
`z_c = 2.33`

The rejection region for this right tailed test is R = {z : z > 2.33}

Test Statistics:

The z-statistic is computed as:

`z ={(\arrow p_1 - \arrow p_2)/(√( \bar p(1- \bar p)(1/n_1 + 1/n_2)) )`

`z ={(0.4195- 0.2741)/(√( 0.3835(1-0.3835)(1/410 + 1/135)) )`

z = 3.014

Decision about the null hypothesis:

Since it is observed that z = 3.014 >
`z_c = 2.33` ; it is concluded that the null hypothesis is rejected

Using the P-value approach: The P-value p = 0.0013 and since p = 0.0013<0.01 , it is concluded that the null hypothesis is rejected.

Conclusion:

It is concluded that the proportion of workers that say monitoring employee e-mail is unethical is greater than the proportion of bosses.