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.
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.
\[
0.398
Solution
Step 1 ：Given a poll of 512 human resource professionals, where $45.5 \%$ said that body piercings and tattoos were big personal grooming red flags.
Step 2 ：We are asked to 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.
Step 3 ：We use the formula for the confidence interval of a proportion, which is given by \(p \pm Z \cdot SE\), where \(p\) is the sample proportion, \(Z\) is the Z-score corresponding to the desired confidence level, and \(SE\) is the standard error.
Step 4 ：The standard error is calculated as \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(n\) is the sample size.
Step 5 ：Substituting the given values, we get \(SE = \sqrt{\frac{0.455(1-0.455)}{512}} = 0.022\).
Step 6 ：The Z-score for a $99 \%$ confidence level is 2.576. Substituting these values into the confidence interval formula, we get \(0.455 \pm 2.576 \cdot 0.022\), which simplifies to \(0.398
Step 7 ：We are also asked to repeat the calculation using a confidence level of $80 \%$. The Z-score for an $80 \%$ confidence level is 1.282.
Step 8 ：Substituting these values into the confidence interval formula, we get \(0.455 \pm 1.282 \cdot 0.022\), which simplifies to \(0.427
Step 9 ：Final Answer: The 99% confidence interval estimate of the proportion of all human resource professionals believing that body piercings and tattoos are big personal grooming red flags is \(\boxed{0.398
From Solvely APP
Solution
Step 1 ：Given a poll of 512 human resource professionals, where $45.5 \%$ said that body piercings and tattoos were big personal grooming red flags.
Step 2 ：We are asked to 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.
Step 3 ：We use the formula for the confidence interval of a proportion, which is given by \(p \pm Z \cdot SE\), where \(p\) is the sample proportion, \(Z\) is the Z-score corresponding to the desired confidence level, and \(SE\) is the standard error.
Step 4 ：The standard error is calculated as \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(n\) is the sample size.
Step 5 ：Substituting the given values, we get \(SE = \sqrt{\frac{0.455(1-0.455)}{512}} = 0.022\).
Step 6 ：The Z-score for a $99 \%$ confidence level is 2.576. Substituting these values into the confidence interval formula, we get \(0.455 \pm 2.576 \cdot 0.022\), which simplifies to \(0.398 Step 7 ：We are also asked to repeat the calculation using a confidence level of $80 \%$. The Z-score for an $80 \%$ confidence level is 1.282. Step 8 ：Substituting these values into the confidence interval formula, we get \(0.455 \pm 1.282 \cdot 0.022\), which simplifies to \(0.427 Step 9 ：Final Answer: The 99% confidence interval estimate of the proportion of all human resource professionals believing that body piercings and tattoos are big personal grooming red flags is \(\boxed{0.398
