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Explain why the margin of error determines the accuracy with which a sample mean estimates a population mean.

The value of a statistic used to estimate a parameter is called a ______ of the parameter.

A variable of a population has a normal distribution. Suppose that you want to find a confidence interval for the population mean.

a. If you know the population standard deviation, which procedure would you use?

b. If you do not know the population standard deviation, which procedure would you use?

Batting Averages. An issue of Scientific American revealed that batting averages, $x$of major-league baseball players are normally

distributed and have a mean of $0.270$ and a standard deviation of $0.031$ For samples of $20$ batting averages, identify the distribution of each variable.

a. $\frac{\stackrel{¯}{x}-0.270}{0.031/\sqrt{20}}$

b. $\frac{\stackrel{¯}{x}-0.270}{s/\sqrt{20}}$

Two $t$-curves have degrees of freedom $12$ and $20$ , respectively. Which one more closely resembles the standard normal curve? Explain your answer.

In each Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following Lacks.

a. Find a $95\%$confidence interval for the population mean. (Note: You may want to review Example $8.2$, which begins on the page $316$)

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

$\hat{x}=35,n=25,\mathrm{σ}=4$

Wedding Costs. According to Bride's Magazine, getting married these days can be expensive when the costs of the reception, engagement ring, bridal gown, and pictures-just to name a few-are included. A simple random sample of $20$ recent U.S. weddings yielded the following data on wedding costs, in dollars.

a. Use the data to obtain a point estimate for the population mean wedding cost. $\mathrm{μ}$, of all recent U.S. weddings. (Note: The sum of the data is $\$526,538$.)

b. Is your point estimate in part (a) likely to equal $\mathrm{μ}$exactly? Explain your answer.

Cottonmouth Litter Size. In the article "The Eastern Cottonmouth (Agkistrodon piscivorus) at the Northern Edge of Its Range" (Journal of Herpetology, Vol. $29$ No. $3$ pp. $391-398$), C. Blem and L. Blem examined the reproductive characteristics of the eastern cottonmouth, a once widely distributed snake whose numbers have decreased recently due to encroachment by humans. A simple random sample of $44$ female cottonmouths yielded the following data on the number of young per litter.

a. Use the data to obtain a point estimate for the mean number of young per litter, $\mathrm{μ}$ of all female eastern cottonmouths. (Note: $\left.\mathrm{Î\pounds }{x}_i=334.\right)$

b. Is your point estimate in part (a) likely to equal $\mathrm{μ}$ exactly? Explain your answer.

What is a confidence interval estimate of a parameter? Why is such an estimate superior to a point estimate?

Fuel Tank Capacity. Consumer Reports provides information on new automobiles models-including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of $35$ new models yielded the following data on fuel tank capacity, in gallons.

a. Determine a point estimate for the mean fuel tank capacity of all new automobile models. Interpret your answer in words. (Note: $\mathrm{Î\pounds }{x}_i=664.9$ gallons.)

b. Determine a $95\%$ confidence interval for the mean fuel tank capacity of all new automobile models. Assume $\mathrm{σ}=3.50$ gallons.

c. How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed?

d. Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer.