Questions: In a poll of 512 human resource professionals, 45.5% said that body piercings and tattoos were big personal grooming red flags. Complete parts (a) through (d) below. a. Among the 512 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. < p < (Round to three decimal places as needed.)

Solution Steps

In a poll of 512 human resource professionals, \( 45.5\% \) 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:

\[ \text{Number of professionals} = 512 \times 0.455 = 233.36 \approx 233 \]

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

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:

• \( \hat{p} = 0.455 \) (sample proportion)
• \( n = 512 \) (sample size)
• \( z \) for \( 99\% \) confidence level is approximately \( 2.576 \).

Calculating the margin of error:

\[ \text{Margin of Error} = 2.576 \cdot \sqrt{\frac{0.455(1 - 0.455)}{512}} = 0.057 \]

Thus, the confidence interval is:

\[ 0.455 \pm 0.057 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 0.455 - 0.057 = 0.398 \] \[ \text{Upper Bound} = 0.455 + 0.057 = 0.512 \]

Therefore, the \( 99\% \) confidence interval for the proportion is \( (0.398, 0.512) \).

Final Answer