Question

It is thought that not as many Americans buy presents to celebrate Valentine’s Day anymore. A random sample of 4000 Americans yielded 2200 who bought their significant other a present and celebrated Valentine’s Day. What size sample should be obtained if a 90% confidence interval was desired to estimate the true proportion within 4%?

a) 625
b) 1000
c) 2400
d) 2500

Answer 1

To estimate the true proportion within 4% for a 90% confidence interval, the formula for sample size estimation of proportions is used. The computed sample size, with the given confidence level and margin of error, is closest to 625 (option a).

Step-by-step explanation:

To determine the sample size needed for a 90% confidence interval to estimate the true proportion within 4%, we use the formula for sample size estimation of proportions: n =[ (Z^2 * p * (1-p))/E^2 ], where Z is the Z-value corresponding to the confidence level, p is the sample proportion, and E is the desired margin of error. From the question, p is estimated using the previous sample as 2200/4000 = 0.55, and E is 0.04. The Z-value for 90% confidence is approximately 1.645. Plugging these values into the formula gives

n = [(1.645^2 * 0.55 * (1-0.55)) / 0.04^2]

This calculation yields a required sample size n, which can be rounded to the nearest whole number. The closest answer from the options provided by the student is 625 (option a). This result indicates that to have a 90% confidence interval with a margin of error of 4%, a random sample of approximately 625 Americans should be obtained.