Question

in a poll of 520 human resource professionals, 45.8% said that body piercings and tattoos were big personal grooming red flags. complete parts (a) and (b) below.
a. among the 520 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags?
(round to the nearest integer as needed.)
b. construct a 99% confidence interval estimate of the proportion of all human resource professionals believing that body piercings and tattoos are big personal grooming red flags.
< p <
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the number of professionals for part (a)

We know that the formula to find the number of people with a certain opinion is \(n = N\times p\), where \(N = 520\) (total number of professionals) and \(p=0.458\) (proportion of professionals with the opinion).

$$n=520\times0.458 = 238.16$$

Rounding to the nearest integer, we get \(n = 238\).

Step2: Calculate the confidence interval for part (b)

For a proportion, the confidence interval formula is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
where \(\hat{p}=0.458\), \(n = 520\), and for a \(99\%\) confidence interval, \(z = 2.576\)
First, calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.458\times(1 - 0.458)}{520}}=\sqrt{\frac{0.458\times0.542}{520}}\)

$$=\sqrt{\frac{0.248236}{520}}\approx\sqrt{0.000477377}\approx0.02185$$

Then, \(z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\times0.02185\approx0.0563\)
The lower limit of the confidence interval is \(\hat{p}-z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.458 - 0.0563=0.402\)
The upper limit of the confidence interval is \(\hat{p}+z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.458+ 0.0563 = 0.514\)

Answer:

a. \(238\)
b. \(0.402 < p < 0.514\)