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Paint Durability. A paint manufacturer in Pittsburgh claims that his paint will last an average of $5$ years. Assuming that paint life is normally distributed and has a standard deviation of $0.5$ year. answer the following questions:

a. Suppose that you paint one house with the paint and that the paint lasts $4.5$ years. Would you consider that evidence against the manufacturer's claim? (Hint: Assuming that the manufacturer's claim is correct, determine the probability that the paint life for a randomly selected house painted with the paint is $4.5$ years or less.)

b. Suppose that you paint $10$ houses with the paint and that the paint lasts an average of $4.5$ years for the $10$ houses. Would you consider that evidence against the manufacturer's claim?

c. Repeat part (b) if the paint lasts an average of $4.9$ years for the $10$ houses painted.

What is the sampling distribution of a statistic? Why is it important?

7.1 Why is sampling often preferable to conducting a census for the purpose of obtaining information about a population?

The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat, One possible starting lineup for that team is as follows:

Part (a): Find the population mean height of the five players.

Part (b): For samples of size 2, construct a table similar to Table 7.2 on page 293. Use the letter in parentheses after each player's name to represent each player.

Part (c): Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.

Part (d): For a random sample of size2, what is the chance that the sample mean will equal the population mean?

Part (e): For a random sample of size 2, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be1 inch or less; that is, determine the probability that $\overline{x}$ will be within1 inch of $\mathrm{μ}$. Interpret your result in terms of percentages.

Each years, Forbers magazine publishes a list of the richest people in the United States. As of September 16, 2013,the six richest Americans and their wealth (to the nearest billion dollars) are as shown in the following table. Consider these six people a population of interest.

Part (a): Calculate the mean wealth, $\mathrm{μ}$, of the six people.

Part (b): For samples of size 2, construct a table similar to Table 7.2 on page 293. (There are 15 possible samples of size 2.)

Part (c): Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.

Part (d): For a random sample of size2, what is the chance that the sample mean will equal the population mean?

Part (e): For a random sample of size 2, determine the probability that the mean wealth of the two people obtained will be within 3 of the population mean. Interpret your result in terms of percentages.

7.2 Why should you generally expect some error when estimating a parameter (e.g., a population mean) by a statistic (e.g., a sample mean)? What is this kind of error called?

America's Riches. Each year, Forbes magazine publishes a list of the richest people in the United States. As of September l6, 2013, the six richest Americans and their wealth (to the neatest billion dollars) are as shown in the following table. Consider these six people a population of interest.

(a) For sample size of $5$construct a table similar to table 7.2 on page293.(There are 6 possible sample) of size $5$

(b) For a random sample of size $5$determine the probability that themean wealth of the two people obtained will be within $3$(i.e,$3$billion) of the population mean. interpret your result in terms of percentages.

Although, in general, you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it?

Why is obtaining the mean and standard deviation of $\stackrel{¯}{x}$ a first step in approximating the sample distribution of the sample mean by a normal distribution?

Does the sample size have an effect on the mean of all possible sample means? Explain your answer.