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Chapter 8: Problem 125

When calculating a confidence interval for the population mean \(\mu\) with a known population standard deviation \(\sigma\), describe the effects of the following two changes on the confidence interval: (1) doubling the sample size, (2) quadrupling (multiplying by 4) the sample size. Give two reasons why this relationship does not hold true if you are calculating a confidence interval for the population mean \(\mu\) with an unknown population standard deviation.

Short Answer

Expert verified
If the population standard deviation is known, doubling or quadrupling the sample size will narrow the confidence interval because it reduces the standard error of the mean. However, this relationship doesn鈥檛 hold if the population standard deviation is unknown because the added uncertainty from estimating it from the sample and the use of the t-distribution (which doesn't narrow as much with increasing sample size as the normal distribution does) can offset the benefits of a larger sample size.

Step by step solution

01

Examine change on confidence interval when sample size is doubled or quadrupled

When the sample size is doubled (n=2n) or quadrupled (n=4n), the standard error of the mean (SEM, represented as \(\sigma / \sqrt{n}\)) decreases, because the denominator of the SEM formula (\(\sqrt{n}\)) gets larger. Since the SEM is in the denominator of the formula for the confidence interval, a smaller SEM would mean a narrower confidence interval. Thus, when the sample size is doubled or quadrupled, the confidence interval becomes narrower, indicating a more precise estimate of the population mean \(\mu\).
02

Explain the reason why this relationship does not hold true if the population standard deviation is unknown

When the population standard deviation is unknown: \n1. We have to estimate it from the sample, which introduces extra uncertainty. This uncertainty isn't reduced by simply increasing sample size. \n2. When population standard deviation is unknown, we use student's t-distribution to calculate the confidence intervals, not the normal distribution. The widths of confidence intervals based on the t-distribution do not decrease at the same rate with increasing sample size as they do with the normal distribution.
03

Summary of the effects

Doubling or quadrupling the sample size narrows the confidence interval when the population standard deviation is known. When the population standard deviation isn't known, these changes in sample size do not cause the same rate of decrease in the width of the confidence interval because: \n1. The added uncertainty from estimating the population standard deviation from the sample. \n2. The reliance on the t-distribution, which does not narrow as much with increasing sample size as the normal distribution does.

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Most popular questions from this chapter

\(8.104\) A random sample of 25 life insurance policyholders showed that the average premium they pay on their life insurance policies is \(\$ 685\) per year with a standard deviation of \(\$ 74\). Assuming that the life insurance policy premiums for all life insurance policyholders have a normal distribution, make a \(99 \%\) confidence interval for the population mean, \(\mu\).

The high cost of health care is a matter of major concern for a large number of families. A random sample of 25 families selected from an area showed that they spend an average of \(\$ 253\) per month on health care with a standard deviation of \(\$ 47 .\) Make a \(98 \%\) confidence interval for the mean health care expenditure per month incurred by all families in this area. Assume that the monthly health care expenditures of all families in this area have a normal distribution.

What assumption(s) must hold true to use the normal distribution to make a confidence interval for the population proportion, \(p\) ?

An economist wants to find a \(90 \%\) confidence interval for the mean sale price of houses in a state. How large a sample should she select so that the estimate is within \(\$ 3500\) of the population mean? Assume that the standard deviation for the sale prices of all houses in this state is \(\$ 31,500\)

At the end of Section \(8.3\), we noted that we always round up when calculating the minimum sample size for a confidence interval for \(\mu\) with a specified margin of error and confidence level. Using the formula for the margin of error, explain why we must always round up in this situation.
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