For part B, construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. b2. In a survey of 2241 U.S. adults in a recent year, 650 made a New Year’s resolution to eat healthier.ÂÂ
5B. For part B, construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.
b2. In a survey of 2241 U.S. adults in a recent year, 650 made a New Year’s resolution to eat healthier.ÂÂ
5C. For part C, construct a 99% confidence interval for the population proportion. Interpret the results.ÂÂ
c1. In a survey of 1000 U.S. adults, 700 think police officers should be required to wear body cameras while on duty.ÂÂ
c2. In a survey of 600 United Kingdom teachers, 226 say they would wear a body camera in school.ÂÂ
5d. You wish to estimate, with 95% confidence, the population proportion of U.S. adults who think congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.ÂÂ
d1. No preliminary estimate is available. Find the minimum sample size needed.ÂÂ
d2. Find the minimum sample size needed, using a prior survey that found that 25% of U.S. adults think congress is doing a good or excellent job.ÂÂ
dc. Compare the results from parts d1 and d2.ÂÂ
5E. In a survey of 1000 U.S. adults, 490 oppose allowing transgender students to use the bathrooms of the opposite biological sex. Construct a 90% confidence interval for the population proportion of u>s. adults who oppose allowing transgender students to use the bathrooms of the opposite biological sex.ÂÂ
The Correct Answer and Explanation is :
The confidence intervals and minimum sample size calculations are as follows:
- Part B: Confidence Intervals for Population Proportions
- 90% Confidence Interval for b2: (0.274, 0.306)
- 95% Confidence Interval for b2: (0.271, 0.309) Interpretation: With 90% confidence, the true proportion of U.S. adults who made a resolution to eat healthier lies between 27.4% and 30.6%. With 95% confidence, this range is slightly wider, from 27.1% to 30.9%. The 95% confidence interval is broader because it represents a higher level of confidence.
- Part C: Confidence Interval for Population Proportion
- 99% Confidence Interval for c1: (0.663, 0.737)
- 99% Confidence Interval for c2: (0.326, 0.428) Interpretation:
- For c1, with 99% confidence, the proportion of U.S. adults who think police officers should wear body cameras is between 66.3% and 73.7%.
- For c2, with 99% confidence, the proportion of UK teachers who would wear body cameras is between 32.6% and 42.8%. These intervals are wider due to the higher confidence level.
- Part E: Confidence Interval
- 90% Confidence Interval for e: (0.464, 0.516) Interpretation: With 90% confidence, the true proportion of U.S. adults who oppose allowing transgender students to use bathrooms of the opposite biological sex is between 46.4% and 51.6%.
- Part D: Minimum Sample Size Calculation
- Minimum Sample Size for d1 (No preliminary estimate): 601
- Minimum Sample Size for d2 (Using prior estimate of 25%): 600 Interpretation: The sample size needed is approximately the same for both parts because using a prior estimate of 25% provides a less variable estimate than assuming no prior estimate.
These results reflect how different levels of confidence (90%, 95%, 99%) influence the width of confidence intervals, with higher confidence intervals being wider to account for more certainty in the estimation.
The table with these values is now displayed for you.