Question

in a poll of 510 human resource professionals, 45% said that body piercings and tattoos were big personal grooming red flags. complete parts (a) and (b) below.

a. among the 510 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 95% 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 professionals for part (a)

To find the number of human - resource professionals who said body piercings and tattoos were big personal - grooming red flags, we use the formula \(n = N\times p\), where \(N = 510\) (the total number of professionals surveyed) and \(p=0.45\) (the proportion of professionals with the given opinion).

$$n = 510\times0.45=229.5\approx230$$

Step2: Calculate the confidence interval for part (b)

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

• Given \(\hat{p}=0.45\), \(n = 510\), and for a \(90\%\) confidence interval, the critical value \(z\) (from the standard normal distribution) is \(z = 1.645\)
• First, calculate the standard error \(SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.45\times(1 - 0.45)}{510}}=\sqrt{\frac{0.45\times0.55}{510}}=\sqrt{\frac{0.2475}{510}}\approx\sqrt{0.0004853}\approx0.022\)
• Then, the margin of error \(ME = z\times SE=1.645\times0.022\approx0.036\)
• The lower limit of the confidence interval is \(\hat{p}-ME=0.45 - 0.036=0.414\)
• The upper limit of the confidence interval is \(\hat{p}+ME=0.45+0.036 = 0.486\)

Answer:

a. \(230\)
b. \(0.414\lt p\lt0.486\)