Questions: The CEO of a chain restaurant wants to estimate the proportion of curbside orders that are handled in five minutes or less. The CEO originally planned to obtain a random sample of 2,000 orders, but is actually able to obtain a sample of 2,500 orders. Which of the following describes the effect of increasing the number of people in the sample? A The standard deviation of the sampling distribution of sample proportions will stay the same. B The standard deviation of the sample proportion will decrease. C The standard deviation of the sampling distribution of sample proportions will increase. D The standard deviation of the population proportion will increase. E The standard deviation of the sampling distribution of sample proportions will decrease.

Solution Steps

The CEO wants to estimate the proportion of curbside orders handled in five minutes or less. The sample size increases from 2,000 to 2,500. We need to determine the effect of this increase on the standard deviation of the sampling distribution of sample proportions.

The standard deviation of the sampling distribution of sample proportions (\( \sigma_{\hat{p}} \)) is given by: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion and \( n \) is the sample size.

As the sample size \( n \) increases, the denominator in the formula \( \sqrt{\frac{p(1-p)}{n}} \) increases. This causes the standard deviation \( \sigma_{\hat{p}} \) to decrease. Therefore, increasing the sample size reduces the standard deviation of the sampling distribution of sample proportions.

From the analysis, the correct statement is that the standard deviation of the sampling distribution of sample proportions will decrease. This corresponds to option E.

Final Answer