Question

in a poll of 515 human resource professionals, 45.8% said that body piercings and tattoos were big personal grooming red flags. complete parts (a) through (d) below
236 (round to the nearest integer as needed.)
a. among the 515 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags?
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 people

We know that the total number of human - resource professionals polled is \(n = 515\), and the proportion \(p=0.458\). To find the number of people who said that body piercings and tattoos were big personal grooming red flags, we use the formula \(x=np\).

$$x = 515\times0.458$$

$$x=515\times\frac{458}{1000}=\frac{515\times458}{1000}$$

$$515\times458=(500 + 15)\times458=500\times458+15\times458=229000+6870 = 235870$$

$$x=\frac{235870}{1000}=235.87\approx236$$

Step2: Calculate the confidence interval

The formula for the confidence interval for a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
For a \(99\%\) confidence interval, the \(z\) - value \(z = 2.576\) (from the standard normal distribution table), \(\hat{p}=0.458\), and \(n = 515\)
First, calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)

$$1-\hat{p}=1 - 0.458=0.542$$

$$\frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.458\times0.542}{515}$$

$$0.458\times0.542=(0.4 + 0.05+0.008)\times0.542=0.4\times0.542+0.05\times0.542+0.008\times0.542$$

$$=0.2168+0.0271+0.004336 = 0.248236$$

$$\frac{0.248236}{515}\approx0.000482$$

$$\sqrt{0.000482}\approx0.022$$

Then, \(z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\times0.022\approx0.057\)
The confidence interval is \(\hat{p}-z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) to \(\hat{p}+z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)

$$0.458-0.057 = 0.401$$

$$0.458 + 0.057=0.515$$

Answer:

a. \(236\)
b. \(0.401