Question

Suppose the CPA Practice Advisor reports that the mean preparation fee for 2017 federal income tax returns was $273. Use this price as the population mean and assume the population standard deviation of preparation fees is $100. If you want the probability that the mean price of a random sample of federal income tax returns is within $16 of the population mean to be at least 0.95 , what is the smallest possible sample size you should collect? Round your answer to the nearest integer.

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Answer 1

The smallest possible sample size needed to ensure a 95% confidence level and a margin of error of $16 is 154.

Step-by-step explanation:

To determine the smallest possible sample size needed to ensure that the probability of the mean price of a random sample of federal income tax returns being within $16 of the population mean is at least 0.95, we can use the formula for the margin of error. The margin of error is found by multiplying the critical value (Z-value) by the population standard deviation divided by the square root of the sample size. In this case, we can use the Z-value corresponding to a 95% confidence level, which is 1.96. So the formula becomes:

MARGIN OF ERROR = (Z-value) * (population standard deviation / square root of sample size)

Given that the population standard deviation is $100 and we want the margin of error to be $16, we can rearrange the formula to solve for the sample size:

sample size ≥ ((Z-value) * population standard deviation / margin of error)^2

Plugging in the values, we get:

sample size ≥ ((1.96) * 100 / 16)^2 = 153.06

Rounding up to the nearest integer, the smallest possible sample size is 154.