Questions: In a poll of 514 human resource professionals, 45.7% said that body piercings and tattoos were big personal grooming red flags. Complete parts (a) through (d) below. a. Among the 514 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags? 235 (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)

Solution Steps

In a poll of 514 human resource professionals, \( 45.7\% \) indicated that body piercings and tattoos were significant personal grooming red flags. To find the number of professionals who expressed this opinion, we calculate:

\[ \text{Number of HR professionals} = 514 \times 0.457 = 235.078 \approx 235 \]

Thus, the number of HR professionals who said that body piercings and tattoos were big personal grooming red flags is \( \boxed{235} \).

To construct a \( 99\% \) confidence interval for the proportion of HR professionals who believe that body piercings and tattoos are significant red flags, we use the formula for the confidence interval for a proportion:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:

• \( \hat{p} = 0.457 \) (sample proportion)
• \( n = 514 \) (sample size)
• \( z \) is the z-value corresponding to the \( 99\% \) confidence level, which is approximately \( 2.576 \).

Calculating the margin of error:

\[ \text{Margin of Error} = z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = 2.576 \cdot \sqrt{\frac{0.457(1 - 0.457)}{514}} \approx 0.057 \]

Thus, the confidence interval is:

\[ 0.457 \pm 0.057 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 0.457 - 0.057 = 0.400 \] \[ \text{Upper Bound} = 0.457 + 0.057 = 0.514 \]

Therefore, the \( 99\% \) confidence interval for the proportion of HR professionals believing that body piercings and tattoos are significant personal grooming red flags is \( (0.400, 0.514) \).

Final Answer