Reading A recent Gallup poll asked Americans to disclose the n...

Question

Reading A recent Gallup poll asked Americans to disclose the number of books they read during the previous year. Initial survey results indicate that s = 16.6 books.

d. How many subjects are needed to estimate the number of books Americans read the previous year within four books with 99% confidence? Compare this result to part (a). How does increasing the level of confidence in the estimate affect sample size? Why is this reasonable?

Accepted Answer

Welcome back, everyone. A nutritionist wants to estimate the average daily calcium intake in milligrams among teenagers in the city. Previous studies report the population standard deviation as sigma equals 120 mg. What is the minimum sample size required to estimate the true mean within + or minus 30 mg with 99% confidence? Additionally, explain briefly why increasing the confidence level from 95% to 99% increases their required sample size, assuming the same margin of error and population standard deviation. For this problem, we're going to identify the minimum sample size using the formula N is greater than or equal to Z multiplied by sigma divided by E2. Now Z is the critical Z value. What we're going to use is the Z 4. The 99% confidence and according to the tables, it is 2.576. We're multiplying by sigma, which is 120, and dividing by the margin of error, which is 30. We're going to square this fraction and we're going to get 106.2. Let's remember that we're going to round to the next integer. So the minimum sample size required is 107, meaning we can cross out options A and C because they have N of 112 and N of 98. So it is either option B or D. So now, if we keep the same margin of error and the same population standard deviation, notice that with an increase in Z and is going to increase, right? And remember that Z increases when confidence increases. So basically, the correct answer for this problem is option B. And is equal to 107. This is the minimum N because, and the answer to the second part of the problem is because of the higher Z value for 99% confidence level, right? With an increase in confidence level, the Z value increases, and because it is in the numerator of the fraction, it is directly proportional to the minimum sample size. Thank you for watching.

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