Final answer:
To estimate the value of the population proportion of those who preferred chocolate ice cream, divide the number who preferred chocolate ice cream by the total number of students. The standard error of the proportion can be calculated using the estimated population proportion and the sample size. A 90% confidence interval can be determined using the estimated population proportion, the standard error, and the Z-score for the desired confidence level.
Step-by-step explanation:
To estimate the value of the population proportion of those who preferred chocolate ice cream, we can use the formula:
Estimated population proportion = (Number of students who preferred chocolate ice cream) / (Total number of students in the sample)
So in this case, the estimated population proportion is:
(175 / 300) = 0.583
The standard error of the proportion is given by the formula:
Standard error of the proportion = sqrt((estimated population proportion * (1 - estimated population proportion)) / n)
Where 'n' is the sample size. In this case, the standard error of the proportion is:
sqrt((0.583 * (1 - 0.583)) / 300) = 0.025
To determine a 90% confidence interval for the population proportion, we can use the formula:
Confidence interval = estimated population proportion ± (Z * standard error of the proportion)
Where 'Z' is the Z-score corresponding to the desired confidence level. For a 90% confidence interval, Z = 1.645. Therefore, the confidence interval is:
0.583 ± (1.645 * 0.025) = (0.537, 0.629)
A 95% confidence interval means that if we were to repeat the sampling process many times and construct 95% confidence intervals, approximately 95% of those intervals would contain the true population proportion. In this study, the 90% confidence interval suggests that we can be 90% confident that the true proportion of students who prefer chocolate ice cream is between 0.537 and 0.629.
If 300 such intervals were determined, the population proportion would be included in about 90% of the intervals.