To calculate the confidence intervals for sample proportions, use the relevant z-score with the formula p ± z \(\sqrt{\frac{p(1-p)}{n}}\). The 95% confidence interval for the city residents owning an automobile is (0.5165, 0.6035). Use similarly calculated intervals for 68% and 99.7% by substituting the appropriate z-scores.
To calculate the confidence intervals for a sample proportion, you can utilize the formula p ± z \(\sqrt{\frac{p(1-p)}{n}}\), where p is the sample proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size. For the given proportion of auto ownership in a city (280 out of 500 residents), the sample proportion (p') is 280/500 = 0.56, and the sample size (n) is 500.
The sample standard deviation (q') can be calculated as \(\sqrt{p'(1-p')}\), which is \(\sqrt{0.56(1-0.56)}\). The z-scores for common confidence levels are approximately 1.00 for 68%, 1.96 for 95%, and 2.575 for 99.7% confidence intervals.
To find the 95% confidence interval, you would use the 1.96 z-score:
0.56 ± 1.96 * \(\sqrt{\frac{0.56(1-0.56)}{500}}\)
To find the 68% and 99.7% confidence intervals, substitute the respective z-scores in place of 1.96.
The calculated 95% confidence interval for this data is (0.5165, 0.6035).