Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals
Answered question
2021-12-16
Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. In a survey of 2149 U.S. adults in a recent year, 1241 made a New Year's resolution to eat healthier.
Answer & Explanation
nick1337
Expert2023-04-21Added 777 answers
To construct a confidence interval for the population proportion, we use the formula:
where:
CI = Confidence Interval
p = sample proportion
= z-score associated with the level of confidence
n = sample size
First, we calculate the sample proportion:
For a 90% confidence interval, the z-score is:
Plugging in the values, we get:
Therefore, we are 90% confident that the true proportion of U.S. adults who made a New Year's resolution to eat healthier is between 0.5476 and 0.6058.
For a 95% confidence interval, the z-score is:
Plugging in the values, we get:
to 0.6151
Therefore, we are 95% confident that the true proportion of U.S. adults who made a New Year's resolution to eat healthier is between 0.5383 and 0.6151.
Comparing the widths of the confidence intervals, we can see that the 95% confidence interval is wider than the 90% confidence interval. This is because a higher level of confidence requires a larger range of possible values to be included in the interval, which results in a wider interval. However, the wider interval also gives us more certainty about the true population proportion.
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