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 only afford to survey 250 customers. How will this decrease in sample size affect the margin of error and confidence interval? True Statements: 1. The margin of error will be about twice as large. 2. The width of the confidence interval will be about twice as large. 3. The margin of error will be about half as large. 4. The width of the confidence interval will be about four times as large. False Statements: 1. The margin of error will be about one-fourth as large. 2. The confidence interval will be narrower. 3. The confidence interval will be wider.
The Correct Answer and Explanation is:
✅ Correct Statements:
- The margin of error will be about twice as large.
- The width of the confidence interval will be about twice as large.
- The confidence interval will be wider.
❌ False Statements:
- The margin of error will be about one-fourth as large.
- The confidence interval will be narrower.
- The margin of error will be about half as large.
- The width of the confidence interval will be about four times as large.
Explanation
In statistics, the margin of error (MOE) in estimating a population proportion is inversely proportional to the square root of the sample size (n). This means that if you reduce the sample size, the margin of error increases. Mathematically, the MOE is proportional to:MOE∝1n\text{MOE} \propto \frac{1}{\sqrt{n}}MOE∝n1
Initially, the sample size was 1000. Due to budget constraints, it’s reduced to 250. That’s a factor of:1000250=4\frac{1000}{250} = 42501000=4
Taking the square root:4=2\sqrt{4} = 24=2
So, the margin of error will approximately double. Since the confidence interval (CI) is constructed by adding and subtracting the margin of error from the sample proportion, the width of the confidence interval is essentially twice the margin of error. Therefore, if the margin of error doubles, the width of the CI also doubles, making the confidence interval wider, not narrower.
This rules out any claim that the margin of error or confidence interval will shrink. The statements saying the margin of error will be one-fourth or one-half as large are incorrect because the sample size is shrinking, not increasing. Also, the claim that the width will be four times as large is too extreme—since it only doubles due to the square root relationship.
In summary, reducing the sample size from 1000 to 250 increases the margin of error and makes the confidence interval wider, reducing the precision of the estimate.
