Chapter 7, Sampling Distributions and the Central Limit Theorem Video Solutions, Mathematical Statistics with Applications

Problem 1

We derived the mean and variance of the random variable $Y$ based on a sample of size 3 from a familiar population, the one associated with tossing a balanced die. Recall that if $Y$ denotes the number of spots observed on the upper face on a single toss of a balanced die, as in Exercise 3.22
$$ \begin{aligned}
P(Y=i) &=1 / 6, \quad i=1,2, \ldots, 6, \\
\mu &=E(Y)=3.5 \\
\operatorname{Var}(Y) &=2.9167
\end{aligned} $$
Use the applet DiceSample (at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html) to complete the following.
a. Use the button "Roll One Set" to take a sample of size 3 from the die-tossing population. What value did you obtain for the mean of this sample? Where does this value fall on the histogram? Is the value that you obtained equal to one of the possible values associated with a single toss of a balanced die? Why or why not?
b. Use the button "Roll One Set" again to obtain another sample of size 3 from the die-tossing population. What value did you obtain for the mean of this new sample? Is the value that you obtained equal to the value you obtained in part (a)? Why or why not?
c. Use the button "Roll One Set" eight more times to obtain a total of ten values of the sample mean. Look at the histogram of these ten means. What do you observe? How many different values for the sample mean did you obtain? Were any values observed more than once?
d. Use the button "Roll 10 Sets" until you have obtained and plotted 100 realized values for the sample mean, $Y$. What do you observe about the shape of the histogram of the 100 realized values? Click on the button "Show Stats" to see the mean and standard deviation of the 100 values $\left(\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{100}\right)$ that you observed. How does the average of the 100 values of $\bar{y}_{i}$ 1, 2, ..., 100 compare to $E(Y)$, the expected number of spots on a single toss of a balanced die? (Notice that the mean and standard deviation of $Y$ that you computed in Exercise 3.22 are given on the second line of the "Stat Report" pop-up screen.)
e. How does the standard deviation of the 100 values of $\bar{y}_{i}, i=1,2, \ldots, 100$ compare to the
standard deviation of $Y$ given on the second line of the "Stat Report" pop-up screen?
f. Click the button "Roll 1000 Sets" a few times, observing changes to the histogram as you generate more and more realized values of the sample mean. How does the resulting histogram compare to the graph given in Figure $7.1(\mathrm{a}) ?$

Shu Naito

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Problem 2

Refer to Example 7.1 and Exercise 7.1
a. Use the method of Example 7.1 to find the exact value of $P(\bar{Y}=2)$
b. Refer to the histogram obtained in Exercise $7.1($ d). How does the relative frequency with which you observed $Y=2$ compare to your answer to part (a)?
c. If you were to generate 10,000 values of $Y$, what do you expect to obtain for the relative frequency of observing $Y=2 ?$

Rashmi Sinha

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Problem 3

Refer to Exercise $7.1 .$ Use the applet DiceSample and scroll down to the next part of the screen that corresponds to taking samples of size $n=12$ from the population corresponding to tossing a balanced die.
a. Take a single sample of size $n=12$ by clicking the button "Roll One Set." Use the button "Roll One Set" to generate nine more values of the sample mean. How does the histogram of observed values of the sample mean compare to the histogram observed in Exercise $7.1(c)$ that was based on ten samples each of size $3 ?$
b. Use the button "Roll 10 Sets" nine more times until you have obtained and plotted 100 realized values (each based on a sample of size $n=12$ ) for the sample mean $Y$. Click on the button "Show Stats" to see the mean and standard deviation of the 100 values $\left(\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{100}\right)$ that you observed.
i. How does the average of these 100 values of $\bar{y}_{i}, i=1,2, \ldots, 100$ compare to the average of the 100 values (based on samples of size $n=3$ ) that you obtained in Exercise $7.1(\mathrm{d}) ?$
ii. Divide the standard deviation of the 100 values of $\bar{y}_{i}, i=1,2, \ldots, 100$ based on samples of size 12 that you just obtained by the standard deviation of the 100 values (based on samples of size $n=3$ ) that you obtained in Exercise 7.1 . Why do you expect to get a value close to $1 / 2 ?\left[\text { Hint: } V(\bar{Y})=\sigma^{2} / n .\right]$
c. Click on the button "Toggle Normal." The (green) continuous density function plotted over the
histogram is that of a normal random variable with mean and standard deviation equal to the mean and standard deviation of the 100 values, $\left(\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{100}\right),$ plotted on the histogram. Does this normal distribution appear to reasonably approximate the distribution described by the histogram?

Shu Naito

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Problem 4

The population corresponding to the upper face observed on a single toss of a balanced die is such that all six possible values are equally likely. Would the results analogous to those obtained in Exercises 7.1 and 7.2 be observed if the die was not balanced? Access the applet DiceSample and scroll down to the part of the screen dealing with "Loaded Die."
a. If the die is loaded, the six possible outcomes are not equally likely. What are the probabilities associated with each outcome? Click on the buttons "1 roll," "10 rolls," and/or "1000 rolls" until you have a good idea of the probabilities associated with the values $1,2,3,4,5,$ and $6 .$ What is the general shape of the histogram that you obtained?
b. Click the button "Show Stats" to see the true values of the probabilities of the six possible values. If $Y$ is the random variable denoting the number of spots on the uppermost face, what is the value for $\mu=E(Y) ?$ What is the value of $\sigma$, the standard deviation of $Y$ ? [Hint: These values appear on the "Stat Report" screen.]
c. How many times did you simulate rolling the die in part (a)? How do the mean and standard deviation of the values that you simulated compare to the true values $\mu=E(Y)$ and $\sigma ?$ Simulate 2000 more rolls and answer the same question.
d. Scroll down to the portion of the screen labeled "Rolling 3 Loaded Dice." Click the button "Roll 1000 Sets" until you have generated 3000 observed values for the random variable $Y$.
i. What is the general shape of the simulated sampling distribution that you obtained?
ii. How does the mean of the 3000 values $\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{300}$ compare to the value of $\mu=E(Y)$ computed in part (a)? How does the standard deviation of the 3000 values compare to $\sigma / \sqrt{3} ?$
e. Scroll down to the portion of the screen labeled "Rolling 12 Loaded Dice."
i. In part (ii), you will use the applet to generate 3000 samples of size $12,$ compute the mean of each observed sample, and plot these means on a histogram. Before using the applet, predict the approximate value that you will obtain for the mean and standard deviation of the 3000 values of $\bar{y}$ that you are about to generate.
ii. Use the applet to generate 3000 samples of size 12 and obtain the histogram associated
with the respective sample means, $\bar{y}_{i}, i=1,2, \ldots, 3000 .$ What is the general shape of the simulated sampling distribution that you obtained? Compare the shape of this simulated sampling distribution with the one you obtained in part (d).
iii. Click the button "Show Stats" to observe the mean and standard deviation of the 3000 values $\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{300} .$ How do these values compare to those you predicted in part (i)?

Shu Naito

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Problem 5

What does the sampling distribution of the sample mean look like if samples are taken from an approximately normal distribution? Use the applet Sampling Distribution of the Mean (at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html) to complete the following. The population to be sampled is approximately normally distributed with $\mu=16.50$ and $\sigma=6.03$ (these values are given above the population histogram and denoted $M$ and S, respectively).
a. Use the button "Next Obs" to select a single value from the approximately normal population. Click the button four more times to complete a sample of size $5 .$ What value did you obtain for the mean of this sample? Locate this value on the bottom histogram (the histogram for the values of $Y$ ).
b. Click the button "Reset" to clear the middle graph. Click the button "Next Obs" five more times to obtain another sample of size 5 from the population. What value did you obtain for the mean of this new sample? Is the value that you obtained equal to the value you obtained in part (a)? Why or why not?
c. Use the button "1 Sample" eight more times to obtain a total of ten values of the sample mean. Look at the histogram of these ten means.
i. What do you observe?
ii. How does the mean of these $10 \bar{y}$ -values compare to the population mean $\mu$ ?
d. Use the button "1 Sample" until you have obtained and plotted 25 realized values for the sample mean $\bar{Y}$, each based on a sample of size 5 .
i. What do you observe about the shape of the histogram of the 25 values of $\bar{y}_{i}$
$$
i=1,2, \ldots, 25 ?
$$
ii. How does the value of the standard deviation of the $25 \bar{y}$ values compare with the theoretical value for $\sigma_{Y}$ obtained in Example 5.27 where we showed that, if $Y$ is computed based on a sample of size $n,$ then $V(\bar{Y})=\sigma^{2} / n ?$
e. Click the button "1000 Samples" a few times, observing changes to the histogram as you generate more and more realized values of the sample mean. What do you observe about the shape of the resulting histogram for the simulated sampling distribution of $Y$ ?
f. Click the button "Toggle Normal" to overlay (in green) the normal distribution with the same mean and standard deviation as the set of values of $Y$ that you previously generated. Does this normal distribution appear to be a good approximation to the sampling distribution of $Y ?$

Shu Naito

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Problem 6

What is the effect of the sample size on the sampling distribution of $Y$ ? Use the applet SampleSize to complete the following. As in Exercise 7.5 , the population to be sampled is approximately normally distributed with $\mu=16.50$ and $\sigma=6.03$ (these values are given above the population histogram and denoted $M$ and $S$, respectively).
a. Use the up/down arrows in the left "Sample Size" box to select one of the small sample sizes
that are available and the arrows in the right "Sample Size" box to select a larger sample size.
b. Click the button "1 Sample" a few times. What is similar about the two histograms that you generated? What is different about them?
c. Click the button "1000 Samples" a few times and answer the questions in part(b).
d. Are the means and standard deviations of the two sampling distributions close to the values that you expected? [Hint: $\left.V(\bar{Y})=\sigma^{2} / n .\right]$
e. Click the button "Toggle Normal." What do you observe about the adequacy of the approximating normal distributions?

Shu Naito

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Problem 7

What does the sampling distribution of the sample variance look like if we sample from a population with an approximately normal distribution? Find out using the applet Sampling Distribution of the Variance (Mound Shaped Population) (at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html) to complete the following.
a. Click the button "Next Obs" to take a sample of size 1 from the population with distribution
represented by the top histogram. The value obtained is plotted on the middle histogram. Click four more times to complete a sample of size $5 .$ The value of the sample variance is computed and given above the middle histogram. Is the value of the sample variance equal to the value of the population variance? Does this surprise you?
b. When you completed part (a), the value of the sample variance was also plotted on the lowest histogram. Click the button "Reset" and repeat the process in part (a) to generate a second observed value for the sample variance. Did you obtain the same value as you observed in part (a)? Why or why not?
c. Click the button "1 Sample" a few times. You will observe that different samples lead to different values of the sample variance. Click the button "1000 Samples" a few times to quickly generate a histogram of the observed values of the sample variance (based on samples of size
5). What is the mean of the values of the sample variance that you generated? Is this mean
close to the value of the population variance?
d. In the previous exercises in this section, you obtained simulated sampling distributions for the sample mean. All these sampling distributions were well approximated (for large sample sizes) by a normal distribution. Although the distribution that you obtained is mound-shaped, does the sampling distribution of the sample variance seem to be symmetric (like the normal distribution)?
e. Click the button "Toggle Theory" to overlay the theoretical density function for the sampling distribution of the variance of a sample of size 5 from a normally distributed population. Does the theoretical density provide a reasonable approximation to the values represented in the histogram?
f. Theorem 7.3 , in the next section, states that if a random sample of size $n$ is taken from a normally distributed population, then $(n-1) S^{2} / \sigma^{2}$ has a $\chi^{2}$ distribution with $(n-1)$ degrees of freedom. Does this result seem consistent with what you observed in parts (d) and (e)?

Shu Naito

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Problem 8

What is the effect of the sample size on the sampling distribution of $S^{2} ?$ Use the applet VarianceSize to complete the following. As in some previous exercises, the population to be sampled is approximately normally distributed with $\mu=16.50$ and $\sigma=6.03$
a. What is the value of the population variance $\sigma^{2} ?$
b. Use the up/down arrows in the left "Sample Size" box to select one of the small sample sizes that are available and the arrows in the right "Sample Size" box to select a larger sample size.
i. Click the button "1 Sample" a few times. What is similar about the two histograms that you generated? What is different about them?
ii. Click the button "1000 Samples" a few times and answer the questions in part (i).
ii. Are the means of the two sampling distributions close to the value of the population variance? Which of the two sampling distributions exhibits smaller variability?
iv. Click the button "Toggle Theory" What do you observe about the adequacy of the approximating theoretical distributions?
c. Select sample sizes of 10 and 50 for a new simulation and click the button " 1000 Samples" a few
times
i. Which of the sampling distributions appear to be more similar to a normal distribution?
ii. Refer to Exercise 7.7(5) . In Exercise 7.97 , you will show that, for a large number of degrees of freedom, the $\chi^{2}$ distribution can be approximated by a normal distribution. Does this seem reasonable based on your current simulation?

Shu Naito

Numerade Educator