Question

in a poll of 513 human resource professionals, 45.6% said that body piercings and tattoos were big personal grooming red flags. complete parts (a) and (b) below.
a. among the 513 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.
(round to three decimal places as needed.)

Explanation:

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

The formula is \(n = N\times p\), where \(N = 513\) and \(p=0.456\).
\(n=513\times0.456 = 234.928\approx235\)

Step2: Calculate the confidence interval for part (b)

The formula for the confidence interval of a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)

• \(\hat{p}=0.456\), \(n = 513\)
• For a \(99\%\) confidence interval, \(z = 2.576\)
• First calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.456\times(1 - 0.456)}{513}}=\sqrt{\frac{0.456\times0.544}{513}}\approx\sqrt{\frac{0.248}{513}}\approx0.022\)
• Then the margin of error \(E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\times0.022\approx0.057\)
• The lower limit is \(\hat{p}-E=0.456 - 0.057=0.399\)
• The upper limit is \(\hat{p}+E=0.456+ 0.057 = 0.513\)

Answer:

a. \(235\)
b. \(0.399 < p < 0.513\)