Question

A marketing assistant for a technology firm plans to randomly select 1000 customers to estimate the proportion who are satisfied with the firm's performance. Based on the results of the survey, the assistant will construct a 95% confidence interval for the proportion of all customers who are satisfied. The marketing manager, however, says that the firm can afford to survey only 250 customers. How will this decrease in sample size affect the margin of error and confidence interval? True Statements False Statements The margin of error will be about 2 times larger. The margin of error will be about 4 times larger The margin of error will be about the same size. The margin of error will be about half as farge. The margin of error will be about one-fourth as large. The confidence interval will be wider. The confidence interval will be narrower.

Answer 1

The decrease in sample size from 1000 to 250 will result in a larger margin of error and a wider confidence interval.

The margin of error can be calculated using the formula:

Margin of error = z * √(p * (1 - p) / n)

where z is the z-score for a desired confidence level, p is the estimated proportion of satisfied customers, and n is the sample size. As the sample size decreases, the margin of error increases. So, in this case, the margin of error will be about 4 times larger with a sample size of 250 compared to a sample size of 1000.

The width of the confidence interval is determined by the margin of error and the sample size. As the margin of error increases, the confidence interval becomes wider. So, the confidence interval will be wider with a sample size of 250 compared to a sample size of 1000.

In summary, the decrease in sample size from 1000 to 250 will result in a larger margin of error and a wider confidence interval. None of the options in the list are completely true.