The Sampling Distribution of the Sample Mean

Population data: $3,4,7,8$

Part (a): Find the mean, $\mu$, 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 $0.5$.

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

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?

Population data: $1,2,3$

Part (a): Find the mean, $\mu$, 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 $0.5$.

Population data: $2,5,8$

Part (a): Find the mean, $\mu$ 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  $0.5$.

Population data: $1,2,3,4$.

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

Part (b): For each of the possible sample sizes, construct a table similar to Table $7.2$ on the page $238$ 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 $0.5$.

Population data: $1,2,3,4,5$

Part (a): Find the mean, $\mu$, 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 $0.5$.

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

Part (a): Find the mean, $\mu$, 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 $0.5$.

Population data: $1,2,3,4,5,6$

Part (a): Find the mean, $\mu$, 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 $0.5$.

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

Part (a): Find the mean, $\mu$, 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 $0.5$.

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 size 2, 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 be 1 inch or less; that is, determine the probability that $\overline{x}$ will be within 1 inch of $\mu$. Interpret your result in terms of percentages.

Repeat parts (b)-(e) of Exercise 7.11 for samples of size 1.

Repeat parts (b)-(e) of Exercise 7.11 for samples of size 3.

Repeat parts (b)-(e) of Exercise 7.11 for samples of size 4.

Repeat parts (b)-(e) of Exercise 7.11 for samples of size 5.

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, $\mu$, 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.l on page 293.
c. Construct a graph similar to Fig.7.3 and interpret your results.
d. For each of the possible sample sises. 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 $0.5$.
7.3 Population data: $1, 2, 3$.

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, $\mu$, 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: $2,5,8$.

 Suppose that a random sample of size $1$ is to be taken from a finite population of size $N$.

a. How many possible samples are there?

b. Identify the relationship between the possible sample means and the        possible observations of the variable under consideration.

c. What is the difference between taking a random sample of size $1$ from     a population and selecting a member at random from the population?

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 $4$ construct a table similar to table 7.2 on page293.(There are 15 possible sample of size $4$

(b) For a random sample of size $4$ 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.

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.

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 $6$ construct a table similar to table 7.2 on page293  what is the  relationship between the only possible sample here and the population?

(b) For a random sample of size $6$ 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.

America's Richest. Explain what the dotplots in part (c)  of exercise 7.17-7.22 illustrate about the impact of increasing sample size on sampling error.

Suppose that a sample is to be taken without replacement from a finite population of size  $N$ if the sample size is the same as the population size

(a) How many possible samples are there?

(b) What are the possible sample means?

(c) What is the relationship between the only possible sample and the population

7.16 NBA Champs. This exercise requires that you have done Exercises 7.11-7.15.
a. Draw a graph similar to that shown in Fig. 7.3 on page $294$ for sample sizes of $1,2,3,4$, and $5$.
b. What does your graph in part (a) illustrate about the impact of increasing sample size on sampling error?
c. Construct a table similar to Table 7.4 on page $294$ for some values of your choice.

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, $\mu$, 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 size 2, 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. 

Repeat parts (b)-(e) of Exercise 7.17 for samples of size 1.

Suppose that a sample is to be taken without replacement from a finite population of size N if the sample size is the same as the population size

(a) How many possible samples are there?

(b) What are the possible sample means?

(c) What is the relationship between the only possible sample and the population

Suppose that a random sample of size 1 is to be taken from a finite population of size N.

a. How many possible samples are there?

b. Identify the relationship between the possible sample means and the possible observations of the variable under consideration.

c. What is the difference between taking a random sample of size 1 from a population and selecting a member at random from the population?

Repeat parts (b)-(e) of Exercise 7.17 for samples of size 3. 

Refer to Exercise $7.10$ on page $295$.

a. Use your answers from Exercise $7.10\left(b\right)$ to determine the mean, ${\mu }_i$, of the variable $\hat{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_{i+}$ of the variable $\hat{x}$, using only your answer from Exercise $7.10\left(a\right)$

Explain why increasing the sample size tends to result in a smaller sampling error when a sample means is used to estimate a population mean.

What is another name for the standard deviation of the variable $\overline{x}$ ? What is the reason for that name?

You have seen that the larger the sample size, the smaller the sampling error tends to be in estimating a population means by a sample mean. This fact is reflected mathematically by the formula for the standard deviation of the sample mean: ${\sigma }_i=\sigma /\sqrt{n}$. For a fixed sample size, explain what this formula implies about the relationship between the population standard deviation and sampling error.

Refer to Exercise 7.3 on page 295 .

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

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

7.34 Refer to Exercise 7.4 on page 295.

a. Use your answers from Exercise 7.4(b) to determine the mean, ${\mu }_5$, of the variable $\tilde{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_5$, of the variable $\overrightarrow{x}$, using only your answer from Exercise 7.4(a).

7.35 Refer to Exercise 7.5 on page 295 .

a. Use your answers from Exercise 7.5(b) to determine the mean, ${\mu }_{\mathrm{i}}$. of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_5$, of the variable $\overline{x}$, using only your answer from Exercise 7.5(a).

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 $\overline{x}$ 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 mean? Explain your answer.

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

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 $\overline{x}$ 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. 

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

Refer to Exercise 7.4 on page 295.

a. Use your answers from Exercise 7.4(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.4(a).

Refer to Exercise 7.5 on page 295.

a. Use your answers from Exercise 7.5(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.5(a).

Refer to Exercise 7.6 on page 295.

a. Use your answers from Exercise 7.6(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.6(a).

Refer to Exercise 7.7 on page 295.

a. Use your answers from Exercise 7.7(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.7(a).

Refer to Exercise 7.8 on page 295.

a. Use your answers from Exercise 7.8(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.8(a).