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.
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?
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.
0.401 < p < 0.515
(round to three decimal places as needed.)
c. repeat part (b) using a confidence level of 80%.
<p <
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the number of people who said body piercings and tattoos were big personal grooming red flags

We know that the total number of people polled \(n = 515\) and the proportion \(\hat{p}=0.458\). The formula for the number of people \(x\) is \(x = n\times\hat{p}\).
So, \(x=515\times0.458\)

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Step2: For part (b) - 99% confidence interval

The formula for a confidence interval for a proportion is \(\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
For a 99% confidence interval, \(\alpha=1 - 0.99 = 0.01\), so \(\alpha/2=0.005\). From the standard normal table, \(z_{\alpha/2}=z_{0.005} = 2.576\)
\(\hat{p}=0.458\), \(n = 515\), \(1-\hat{p}=0.542\)
The margin of error \(E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\sqrt{\frac{0.458\times0.542}{515}}\)

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\(E=2.576\times0.022\approx0.057\)
The confidence interval is \(\hat{p}-E\(0.458-0.057\(0.401

Step3: For part (c) - 80% confidence interval

\(\alpha=1 - 0.80=0.20\), \(\alpha/2 = 0.10\). From the standard normal table, \(z_{\alpha/2}=z_{0.10}=1.28\)
The margin of error \(E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.28\sqrt{\frac{0.458\times0.542}{515}}\)
Since \(\sqrt{\frac{0.458\times0.542}{515}}\approx0.022\) (calculated above)
\(E=1.28\times0.022\approx0.028\)
The confidence interval is \(\hat{p}-E\(0.458 - 0.028\(0.430

Answer:

a. \(236\)
b. \(0.401 < p<0.515\)
c. \(0.430 < p<0.486\)