Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.
a. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be $95 \%$ confident that the population proportion is estimated to within 0.03 ?
b. If it were later determined that it was important to be more than $9 \%$ confident and a new survey was commissioned, how would that affect the minimum number of people you would need to survey?
A. Increase
B. Decrease
C. No change
Answer
For the second part of the question, if it were later determined that it was important to be more than 9% confident, the minimum number of people you would need to survey would \(\boxed{Increase}\). This is because a higher confidence level requires a larger sample size to maintain the same margin of error.
Step 1 ：Define the constants for the problem: the z-score for a 95% confidence level (Z) is 1.96, the most conservative estimate of the population proportion (p) is 0.5, and the desired margin of error (E) is 0.03.
Step 2 ：Use the formula for minimum sample size in a proportion estimation: \(n = \lceil \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}} \rceil\).
Step 3 ：Substitute the values into the formula: \(n = \lceil \frac{{1.96^2 \cdot 0.5 \cdot (1-0.5)}}{{0.03^2}} \rceil\).
Step 4 ：Solve the equation to find the minimum sample size, which is \(\boxed{1068}\).
Step 5 ：For the second part of the question, if it were later determined that it was important to be more than 9% confident, the minimum number of people you would need to survey would \(\boxed{Increase}\). This is because a higher confidence level requires a larger sample size to maintain the same margin of error.
