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Chapter 7: Q. 7.29 (page 300)

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

Short Answer

Expert verified

Yes, the sample size has an effect on the standard deviation of all possible sample means.

Step by step solution

01

Step 1. Given Information

We need to identify whether the sample size has an effect on the standard deviation of all possible sample means.

02

Step 2. Explanation

For samples of size n,the standard deviation of the variable x炉equals the standard deviation of the variable under consideration divided by the square root of the sample size. That is, 蟽x炉=蟽n.

So, it can be seen that the standard deviation of all possible sample means depends on the sample size n.

And it can be seen that as the sample size increase the standard deviation ofx炉decreases.

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Most popular questions from this chapter

According to the central limit theorem, for a relatively large sample size, the variable x~is approximately normally distributed.

a. What rule of thumb is used for deciding whether the sample size is relatively large?

b. Roughly speaking, what property of the distribution of the variable under consideration determines how large the sample size must be for a normal distribution to provide an adequate approximation to the distribution of x~ ?

Ethanol Railroad Tariffs. An ethanol railroad tariff is a fee charged for shipments of ethanol on public railroads. The Agricultural Marketing Service publishes tariff rates for railroad-car shipments of ethanol in the Biofuel Transportation Database. Assuming that the standard deviation of such tariff rates is \(1,150, determine the probability that the mean tariff rate of 500randomly selected railroad car shipments of ethanol will be within \)100of the mean tariff rate of all railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.

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.

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?

Why is obtaining the mean and standard deviation of x炉 a first step in approximating the sample distribution of the sample mean by a normal distribution?
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