Question

in a poll of 517 human resource professionals, 45.6% said that body piercings and tattoos were big personal grooming red flags. complete parts (a) through (d) below.
a. among the 517 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags?
236 (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 for part (a)

The number of people is \(n = 517\) and the proportion \(p = 0.456\). The number of people who said the statement is \(n\times p=517\times0.456 = 235.752\approx236\) (rounded to the nearest integer).

Step2: Find the confidence interval for part (b)

For a proportion, the confidence interval formula is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Here, \(\hat{p}=0.456\), \(n = 517\), and for a \(99\%\) confidence interval, \(z = 2.576\).
First, calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.456\times(1 - 0.456)}{517}}=\sqrt{\frac{0.456\times0.544}{517}}=\sqrt{\frac{0.248064}{517}}\approx\sqrt{0.0004798}\approx0.0219\)
Then, \(z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\times0.0219\approx0.0564\)
The lower bound is \(\hat{p}-z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.456- 0.0564=0.3996\)
The upper bound is \(\hat{p}+z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.456 + 0.0564=0.5124\)

Answer:

a. \(236\)
b. \(0.400 < p<0.512\)