Questions: In a poll of 510 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 510 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags? 233 (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. 0.400<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.)

Solution Steps

In a poll of 510 human resource professionals, \( 45.7\% \) indicated that body piercings and tattoos were significant personal grooming red flags. The number of professionals who expressed this opinion can be calculated as follows:

\[ \text{Number} = 510 \times 0.457 = 233 \]

Thus, the number of HR professionals who said that body piercings and tattoos were big personal grooming red flags is \( \boxed{233} \).

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

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

Where:

• \( \hat{p} = 0.457 \)
• \( n = 510 \)
• \( z \) for \( 99\% \) confidence level is approximately \( 2.576 \)

Calculating the margin of error:

\[ \text{Margin of Error} = 2.576 \cdot \sqrt{\frac{0.457(1 - 0.457)}{510}} \approx 0.057 \]

Thus, the confidence interval is:

\[ 0.457 \pm 0.057 \implies (0.400, 0.514) \]

The \( 99\% \) confidence interval is \( \boxed{(0.400, 0.514)} \).

For an \( 80\% \) confidence interval, we again use the same formula, but with \( z \) for \( 80\% \) confidence level, which is approximately \( 1.282 \):

Calculating the margin of error:

\[ \text{Margin of Error} = 1.282 \cdot \sqrt{\frac{0.457(1 - 0.457)}{510}} \approx 0.028 \]

Thus, the confidence interval is:

\[ 0.457 \pm 0.028 \implies (0.429, 0.485) \]

The \( 80\% \) confidence interval is \( \boxed{(0.429, 0.485)} \).

Final Answer