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In a poll of 512 human resource professionals, [tex]$45.7 \%$[/tex] 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?
234 (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.
[tex]$\square \, \square \, \ \textless \ \, p \, \ \textless \ \, \square \, \square$[/tex]
(Round to three decimal places as needed.)



Answer :

GinnyAnswer
Let's go through each part step-by-step:

### Part (a)
To determine how many of the 512 human resource professionals said that body piercings and tattoos were big personal grooming red flags when 45.7% of them believe this, we perform the following calculation:

1. Calculate the number using the given proportion:
[tex]\[ \text{Number of professionals} = \text{Sample size} \times \text{Proportion} \][/tex]
Substituting the values:
[tex]\[ \text{Number of professionals} = 512 \times 0.457 = 234.144 \][/tex]

2. Round to the nearest integer:
[tex]\[ \text{Number of professionals} \approx 234 \][/tex]

So, 234 HR professionals out of 512 said that body piercings and tattoos were big personal grooming red flags.

### Part (b)
To construct a 99% confidence interval for the proportion, we need to calculate the margin of error and then determine the lower and upper bounds of the interval.

1. Proportion ([tex]\(p\)[/tex]):
[tex]\[ p = 0.457 \][/tex]

2. Sample size ([tex]\(n\)[/tex]):
[tex]\[ n = 512 \][/tex]

3. Confidence level: 99%
- The critical value [tex]\(z\)[/tex] for a 99% confidence level is obtained from the standard normal distribution, which is approximately 2.576.

4. Calculate the standard error (SE):
[tex]\[ \text{Standard Error} = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
Substituting values:
[tex]\[ \text{Standard Error} = \sqrt{\frac{0.457 \times (1 - 0.457)}{512}} \approx 0.0223 \][/tex]

5. Calculate the margin of error (ME):
[tex]\[ \text{Margin of Error} = z \times \text{Standard Error} \][/tex]
Substituting values:
[tex]\[ \text{Margin of Error} = 2.576 \times 0.0223 \approx 0.0576 \][/tex]

6. Determine the confidence interval:
- Lower bound:
[tex]\[ \text{Lower bound} = p - \text{Margin of Error} \approx 0.457 - 0.0576 = 0.3994 \rightarrow 0.4 \text{ (rounded to 3 decimal places)} \][/tex]
- Upper bound:
[tex]\[ \text{Upper bound} = p + \text{Margin of Error} \approx 0.457 + 0.0576 = 0.5146 \rightarrow 0.514 \text{ (rounded to 3 decimal places)} \][/tex]

Thus, the 99% confidence interval estimate for the proportion is:
[tex]\[ 0.4 < p < 0.514 \][/tex]

Summarized:

- Part (a): 234 HR professionals.
- Part (b): The 99% confidence interval is [tex]\(0.4 < p < 0.514\)[/tex].

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