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

a)
`p_(w) -p_(b)`

With
`p_(w)` the proportion associated to the workers and
`p_(b)` the proportion for the bosses

b)
`\hat p_(w) -\hat p_(b) \sim N(p_w -p_b, \sqrt{(p_w (1-p_w))/(n_w) +(p_b (1-p_b))/(n_b) }`

With
`\hat p_w = 0.420, p_b = 0.274 , n_w = 410, n_b= 135`

Explanation:

Part a

For this case we define the parameter of interest is the difference between the proportion of workers that say monitoring employee e-mail is unethical and the proportion of bosses that say monitoring employee e-mail is unethical

So the parameter can be expressed like this:

`p_(w) -p_(b)`

With
`p_(w)` the proportion associated to the workers and
`p_(b)` the proportion for the bosses

Part b

For this case we want to find the distribution for
`p_(w) -p_(b)`

The estimated proportions for this case are:

`\hat p_(w)= (172)/(410)= 0.420`

`\hat p_(b)= (37)/(135)= 0.274`

We can assume that these estimators represnent unbiased estimators for the real parameter so then the distribution for the difference in the proportions can be assumed like this:

`\hat p_(w) -\hat p_(b) \sim N(p_w -p_b, \sqrt{(p_w (1-p_w))/(n_w) +(p_b (1-p_b))/(n_b) }`

With
`\hat p_w = 0.420, p_b = 0.274 , n_w = 410, n_b= 135`