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Chapter 7: Q.7.26 (page 300)

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

Short Answer

Expert verified

The sampling distribution of a sample is frequently approximated by the Normal Distribution.

Step by step solution

01

Given Information

We have to find the method to approximate the sample size.

02

Explanation

The variable $x$ is nearly normally distributed, according to the central limit theorem, for a relatively large sample size.

The sampling distribution of a sample is frequently approximated by the Normal Distribution.

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Most popular questions from this chapter

A variable of a population has a mean of $μ = 100$ and a standard deviation of $σ = 28 .$

a. Identify the sampling distribution of the sample mean for samples of size $49 .$

b.In answering part (a), what assumption did you make about the distribution of the variable?

c. Can you answer part (a) if the sample size is $16$ instead of $49 ?$

Why or why not?

Population data: $2, 3, 5, 5, 7, 8$

Part (a): Find the mean, $μ$ , of the variable.

Part (b): For each of the possible sample sizes, construct a table similar to Table $7.2$ on the page $293$ and draw a dotplot for the sampling for the sampling distribution of the sample mean similar to Fig $7.1$ on page $293$ .

Part (c): Construct a graph similar to Fig $7.3$ and interpret your results.

Part (d): For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.

Part (e): For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be $0.5$ or less, that is, that the absolute value of the difference between the sample mean and the population mean is at most $0.5$ .

7.56 Heights of Starting Players. In Example $7.5$ , we used the definition of the standard deviation of a variable (Definition $3.12$ on page $142$ ) to obtain the standard deviation of the heights of the five starting players on a men's basketball team and also the standard deviation of $\bar{x}$ for samples of sizes $1, 2, 3, 4,$ and $5$ . The results are summarized in Table $7.6$ on page $298$ . Because the sampling is without replacement from a finite population, Equation $(7.1)$ can also be used to obtain role="math" localid="1651069065157" $σ_{x}$ .
a. Apply Equation $(7.1)$ to compute role="math" localid="1651069501306" $σ_{\bar{x}}$ for samples of sizes $1, 2, 3, 4,$ and 5. Compare your answers with those in Table $7.6$ .
b. Use the simpler formula, Equation $(7.2)$ , to compute role="math" localid="1651069072557" $σ_{x}$ for samples of sizes $1, 2, 3, 4,$ and $5$ . Compare your answers with those in Table $7.6 .$ Why does Equation $(7.2)$ generally yield such poor approximations to the true values?
c. What percentages of the population size are samples of sizes $1, 2, 3 . 4,$ and $5$ ?

A variable of a population has mean μ and standard deviation $σ$ . that For a large sample size n, answer the following questions.

a. Identify the distribution of $x .$

b. Does your answer to part (a) depend on n being large? Explain your answer.

c. Identify the mean and the standard deviation of $x .$

d. Does your answer to part (c) depend on the sample size being large? Why or why not?

A variable of a population is normally distribution with mean $μ$ and standard deviation $σ .$

a. Identify the distribution of $x .$

b. Does your answer to part (a) depend on the sample size? Explain your answer.

c. Identify the mean and the standard deviation of $x .$

d. Does your answer to part (c) depend on the assumption that the variable under consideration is normally distributed? Why or why not?

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