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Math  /  Data & Statistics

QuestionIn a poll of 510 human resource professionals, 45.9%45.9 \% 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? □\square (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. □\square <p<<p< □\square (Round to three decimal places as needed.) c. Repeat part (b) using a confidence level of 80%80 \%. □\square < <<<< □\square (Round to three decimal places as needed.) d. Compare the confidence intervals from parts (b) and (c) and identify the interval that is wider. Why is it wider? proportion. proportion. proportion. proportion.

Studdy Solution

STEP 1

1. The sample size is n=510 n = 510 .
2. The sample proportion of HR professionals who view body piercings and tattoos as red flags is p^=0.459 \hat{p} = 0.459 .
3. We will use the standard normal distribution for confidence interval calculations.

STEP 2

1. Calculate the number of HR professionals who view body piercings and tattoos as red flags.
2. Construct a 99% confidence interval for the population proportion.
3. Construct an 80% confidence interval for the population proportion.
4. Compare the confidence intervals and determine which is wider.

STEP 3

Calculate the number of HR professionals who said body piercings and tattoos were red flags:
Number=0.459×510 \text{Number} = 0.459 \times 510
Number=234.09 \text{Number} = 234.09
Round to the nearest integer:
234 \boxed{234}

STEP 4

Calculate the standard error for the proportion:
SE=p^(1−p^)n=0.459×(1−0.459)510 SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.459 \times (1-0.459)}{510}}
SE≈0.0219 SE \approx 0.0219
Find the critical value for a 99% confidence interval (using z≈2.576 z \approx 2.576 ):
ME=z×SE=2.576×0.0219 ME = z \times SE = 2.576 \times 0.0219
ME≈0.0564 ME \approx 0.0564
Construct the confidence interval:
p^−ME<p<p^+ME \hat{p} - ME < p < \hat{p} + ME
0.459−0.0564<p<0.459+0.0564 0.459 - 0.0564 < p < 0.459 + 0.0564
0.403<p<0.515 0.403 < p < 0.515
0.403<p<0.515 \boxed{0.403 < p < 0.515}

STEP 5

Find the critical value for an 80% confidence interval (using z≈1.282 z \approx 1.282 ):
ME=z×SE=1.282×0.0219 ME = z \times SE = 1.282 \times 0.0219
ME≈0.0281 ME \approx 0.0281
Construct the confidence interval:
p^−ME<p<p^+ME \hat{p} - ME < p < \hat{p} + ME
0.459−0.0281<p<0.459+0.0281 0.459 - 0.0281 < p < 0.459 + 0.0281
0.431<p<0.487 0.431 < p < 0.487
0.431<p<0.487 \boxed{0.431 < p < 0.487}

STEP 6

Compare the confidence intervals:
- The 99% confidence interval is 0.403<p<0.515 0.403 < p < 0.515 . - The 80% confidence interval is 0.431<p<0.487 0.431 < p < 0.487 .
The 99% confidence interval is wider because a higher confidence level requires a larger margin of error to ensure that the interval captures the true population proportion with greater certainty.

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