91Ó°ÊÓ

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

Refer to Exercise 7.3 on page 295 .

a. Use your answers from Exercise 7.3(b) to determine the mean, ${\mathrm{μ}}_s$. of the variable $\stackrel{¯}{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mathrm{μ}}_i$, of the variable $\tilde{x}$, using only your answer from Exercise 7.3(a).

In Exercises 7.3-7.10, we have given population data for a variable. For each exercise, do the following tasks.
a. Find the mean, $\mathrm{μ}$, of the variable.
b. For each of the possible sample sizes, construct a table similar to Table 7.2 on page 293 and draw a dotplot for the sampling distribution of the sample mean similar to Fig. 7.1 on page 293.
c. Construct a graph similar to Fig. 7.3 and interpret your results.
d. For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.
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 (in magnitude), that is, that the absolute value of the difference between the sample mean and the population mean is at most $0.5$.
7.4 Population data: $2,5,8$.

Baby Weight. The paper "Are Babies Normal?" by T. Clemons and M. Pagano (The American Statistician, Vol. 53, No, 4. pp. 298-302) focused on birth weights of babies. According to the article, the mean birth weight is 3369 grams (7 pounds, 6.5 ounces) with a standard deviation of 581 grams.
a. Identify the population and variable.
b. For samples of size 200, find the mean and standard deviation of all possible sample mean weights.
c. Repeat part (b) for samples of size 400.

A statistic is said to be an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, it is said to be a biased estimator. An unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not.

Part (a): Is the sample mean an unbiased estimator of the population mean? Explain your answer.

Part (b): Is the sample median an unbiased estimator of the population mean? Explain your answer.

A variable of a population has a mean of $\mathrm{μ}=35$and a standard deviation of $\mathrm{σ}=42.$

a. If the variable is normally distributed, identify the sampling distribution of the sample mean for samples of size $9.$

b. Can you answer part (a) if the distribution of the variable under consideration is unknown? Explain your answer.

c. Can you answer part (a) if the distribution of the variable under consideration is unknown but the sample size is $36$instead of $9$?

Why or why not?

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

a. Identify the distribution of$\overline{x}.$

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

c. Identify the mean and the standard deviation of$\overline{x}.$

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

Refer to Fig. $7.6$on page 306 .

a. Why are the four graphs in Fig. 7.6(a) all centered at the same place?

b. Why does the spread of the graphs diminish with increasing sample size? How does this result affect the sampling error when you estimate a population mean, $\mathrm{μ}$by a sample mean, $\tilde{x}$ ?

c. Why are the graphs in Fig. 7.6(a) bell shaped?

d. Why do the graphs in Figs. $7.6\left(b\right)$and (c) become bell shaped as the sample size increases?

According to the central limit theorem, for a relatively large sample size, the variable $\tilde{x}$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 $\tilde{x}$ ?