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) through (d) 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?
238 (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.
0.402 < p < 0.514
(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: Identify the formula for confidence interval of proportion

The formula for the confidence interval of a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z - score corresponding to the confidence level.

Given \(\hat{p}=0.458\), \(n = 520\)

For an \(80\%\) confidence level, the z - score \(z\) (using standard normal distribution tables or a calculator) is \(z=1.28\)

Step2: Calculate the margin of error \(E\)

First, calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.458\times(1 - 0.458)}{520}}=\sqrt{\frac{0.458\times0.542}{520}}\)

\(0.458\times0.542 = 0.248236\)

\(\frac{0.248236}{520}\approx0.0004774\)

\(\sqrt{0.0004774}\approx0.02185\)

Margin of error \(E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.28\times0.02185\approx0.028\)

Step3: Calculate the lower and upper bounds of the confidence interval

Lower bound \(=\hat{p}-E=0.458- 0.028=0.430\)

Upper bound \(=\hat{p}+E=0.458 + 0.028=0.486\)

Answer:

\(0.430