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Modern Mathematical Statistics with Applications

Devore, Jay L., Berk, Kenneth N.

Chapter 6

Statistics and Sampling Distributions - all with Video Answers

Educators

Section 1

Statistics and Their Distributions

02:47

Problem 1

A particular brand of dishwasher soap is sold in three sizes: $25 \mathrm{oz}, 40 \mathrm{oz}$, and $65 \mathrm{oz}$. Twenty percent of all purchasers select a $25-\mathrm{oz}$ box, $50 \%$ select a $40-\mathrm{oz}$ box, and the remaining $30 \%$ choose a $65-\mathrm{oz}$ box. Let $X_{1}$ and $X_{2}$ denote the package sizes selected by two independently selected purchasers.
a. Determine the sampling distribution of $X$, calculate $E(\bar{X})$, and compare to $\mu$.
b. Determine the sampling distribution of the sample variance $S^{2}$, calculate $E\left(S^{2}\right)$, and compare to $\sigma^{2}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:54

Problem 2

There are two traffic lights on the way to work. Let $X_{1}$ be the number of lights that are red, requiring a stop, and suppose that the distribution of $X_{1}$ is as follows:
Let $X_{2}$ be the number of lights that are red on the way home; $X_{2}$ is independent of $X_{1}$. Assume that $X_{2}$ has the same distribution as $X_{1}$, so that $X_{1}$, $X_{2}$ is a random sample of size $n=2$.
a. Let $T_{\mathrm{o}}=X_{1}+X_{2}$, and determine the probability distribution of $T_{\mathrm{o}}$.
b. Calculate $\mu_{T_{0}}$. How does it relate to $\mu$, the population mean?
c. Calculate $\sigma_{T_{0}}^{2}$. How does it relate to $\sigma^{2}$, the population variance?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:35

Problem 3

It is known that $80 \%$ of all brand A DVD players work in a satisfactory manner throughout the warranty period (are "successes"). Suppose that $n=10$ players are randomly selected. Let $X=$ the number of successes in the sample. The statistic $X / n$ is the sample proportion (fraction) of successes. Obtain the sampling distribution of this statistic. [Hint: One possible value of $X / n$ is $.3$, corresponding to $X=3$. What is the probability of this value (what kind of random variable is $X)$ ?]

Trinity Steen
Trinity Steen
Numerade Educator
01:56

Problem 4

A box contains ten sealed envelopes numbered 1 , $\ldots, 10$. The first five contain no money, the next three each contain $\$ 5$, and there is a $\$ 10$ bill in each of the last two. A sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If $X_{1}, X_{2}$, and $X_{3}$ denote the amounts in the selected envelopes, the statistic of interest is $M=$ the maximum of $X_{1}, X_{2}$, and $X_{3}$.
a. Obtain the probability distribution of this statistic.
b. Describe how you would carry out a simulation experiment to compare the distributions of $M$ for various sample sizes. How would you guess the distribution would change as $n$ increases?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:53

Problem 5

Let $X$ be the number of packages being mailed by a randomly selected customer at a shipping facility. Suppose the distribution of $X$ is as follows:
a. Consider a random sample of size $n=2$ (two customers), and let $X$ be the sample mean number of packages shipped. Obtain the probability distribution of $\bar{X}$.
b. Refer to part (a) and calculate $P(X \leq 2.5)$.
c. Again consider a random sample of size $n=2$, but now focus on the statistic $R=$ the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of $R$. [Hint: Calculate the value of $R$ for each outcome and use the probabilities from part (a).]
d. If a random sample of size $n=4$ is selected, what is $P(\bar{X} \leq 1.5)$ ? [Hint: You should not have to list all possible outcomes, only those for which $x \leq 1.5 .]$

Hossam Mohamed
Hossam Mohamed
Numerade Educator
02:00

Problem 6

A company maintains three offices in a region, each staffed by two employees. Information concerning yearly salaries (1000's of dollars) is as follows:
$$
\begin{array}{lcccccc}
\text { Office } & 1 & 1 & 2 & 2 & 3 & 3 \\
\text { Employee } & 1 & 2 & 3 & 4 & 5 & 6 \\
\text { Salary } & 29.7 & 33.6 & 30.2 & 33.6 & 25.8 & 29.7
\end{array}
$$
a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary $\bar{X}$.
b. Suppose one of the three offices is randomly selected. Let $X_{1}$ and $X_{2}$ denote the salaries of the two employees. Determine the sampling distribution of $X$.
c. How does $E(X)$ from parts (a) and (b) compare to the population mean salary $\mu$ ?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:23

Problem 7

The number of dirt specks on a randomly selected square yard of polyethylene film of a certain type has a Poisson distribution with a mean value of 2 specks per square yard. Consider a random sample of $n=5$ film specimens, each having area 1 square yard, and let $X$ be the resulting sample mean number of dirt specks. Obtain the first 21 probabilities in the $\bar{X}$ sampling distribution. [Hint: What does a moment generating function argument say about the distribution of $X_{1}+\cdots+X_{3}$ ?]

Manik Pulyani
Manik Pulyani
Numerade Educator
01:58

Problem 8

Suppose the amount of liquid dispensed by a machine is uniformly distributed with lower limit $A=8 \mathrm{oz}$ and upper limit $B=10 \mathrm{oz}$. Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes $n=5$, 10,20 , and 30 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:32

Problem 9

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of $X$ when the population distribution is Weibull with $\alpha=2$ and $\beta=5$, as in Example 6.1. Consider the four sample sizes $n=5$, 10,20 , and 30 , and in each case use 500 replications. For which of these sample sizes does the $\bar{X}$ sampling distribution appear to be approximately normal?

Dominador Tan
Dominador Tan
Numerade Educator
01:32

Problem 10

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of $X$ when the population distribution is lognormal with $E[\ln (X)]=3$ and $V[\ln (X)]=1$. Consider the four sample sizes $n=10,20,30$, and 50 , and in each case use 500 replications. For which of these sample sizes does the $X$ sampling distribution appear to be approximately normal?

Dominador Tan
Dominador Tan
Numerade Educator
09:15

Problem 11

Of $n_{1}$ randomly selected male smokers, $X_{1}$ smoked filter cigarettes, whereas of $n_{2}$ randomly selected female smokers, $X_{2}$ smoked filter cigarettes. Let $p_{1}$ and $p_{2}$ denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes.
a. Show that $\left(X_{1} / n_{1}\right)-\left(X_{2} / n_{2}\right)$ is an unbiased estimator for $p_{1}-p_{2}$. [Hint: $E\left(X_{i}\right)=n_{i} p_{i}$ for $i=1,2 .]$
b. What is the standard error of the estimator in part (a)?
c. How would you use the observed values $x_{1}$ and $x_{2}$ to estimate the standard error of your estimator?
d. If $n_{1}=n_{2}=200, x_{1}=127$, and $x_{2}=176$, use the estimator of part (a) to obtain an estimate of $p_{1}-p_{2}$.
e. Use the result of part (c) and the data of part (d) to estimate the standard error of the estimator.

Robin Corrigan
Robin Corrigan
Numerade Educator
01:23

Problem 12

Suppose a certain type of fertilizer has an expected yield per acre of $\mu_{1}$ with variance $\sigma^{2}$, whereas the expected yield for a second type of fertilizer is $\mu_{2}$ with the same variance $\sigma^{2}$. Let $S_{1}^{2}$ and $S_{2}^{2}$ denote the sample variances of yields based on sample sizes $n_{1}$ and $n_{2}$, respectively, of the two fertilizers. Show that the pooled (combined) estimator
$$
\partial^{2}=\frac{\left(n_{1}-1\right) S_{1}^{2}+\left(n_{2}-1\right) S_{2}^{2}}{n_{1}+n_{2}-2}
$$
is an unbiased estimator of $\sigma^{2}$.

Clarissa Noh
Clarissa Noh
Numerade Educator
02:23

Problem 13

Consider a random sample $X_{1}, \ldots, X_{n}$ from the pdf
$$
f(x ; \theta)=.5(1+\theta x) \quad-1 \leq x \leq 1
$$
where $-1 \leq \theta \leq 1$ (this distribution arises in particle physics). Show that $\hat{\theta}=3 \bar{X}$ is an unbiased estimator of $\theta$. [Hint: First determine $\mu=E(X)=E(X) .$.

Robin Corrigan
Robin Corrigan
Numerade Educator
02:06

Problem 14

A sample of $n$ captured Pandemonium jet fighters result in serial numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. The CIA knows that the aircraft were numbered consecutively at the factory starting with $\alpha$ and ending with $\beta$, so that the total number of planes manufactured is $\beta-\alpha+1$ (e.g., if $\alpha=17$ and $\beta=29$, then $29-17+1=13$ planes having serial numbers $17,18,19, \ldots, 28,29$ were manufactured). However, the CIA does not know the values of $\alpha$ or $\beta$. A CIA statistician suggests using the esti$\operatorname{mator} \max \left(X_{j}\right)-\min \left(X_{i}\right)+1$ to estimate the total number of planes manufactured.
a. If $n=5, x_{1}=237, x_{2}=375, x_{3}=202$, $x_{4}=525$, and $x_{5}=418$, what is the corresponding estimate?
b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating $\beta$ $\alpha+1$ ? Explain in one or two sentences.
(A similar method was used to estimate German tank production in World War II.)

Clarissa Noh
Clarissa Noh
Numerade Educator
07:54

Problem 15

Let $X_{1}, X_{2}, \ldots, X_{n}$ represent a random sample from a Rayleigh distribution with pdf
$$
f(x ; \theta)=\frac{x}{\theta} e^{-x^{2} /(2 \theta)} \quad x>0
$$
a. It can be shown that $E\left(X^{2}\right)=2 \theta$. Use this fact to construct an unbiased estimator of $\theta$ based on $\sum X_{i}^{2}$ (and use rules of expected value to show that it is unbiased).
b. Estimate $\theta$ from the following measurements of blood plasma beta concentration (in $\mathrm{pmol} / \mathrm{L}$ ) for $n=10$ men.
$\begin{array}{lllll}16.88 & 10.23 & 4.59 & 6.66 & 13.68 \\ 14.23 & 19.87 & 9.40 & 6.51 & 10.95\end{array}$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
02:00

Problem 16

Suppose the true average growth $\mu$ of one type of plant during a l-year period is identical to that of a second type, but the variance of growth for the first type is $\sigma^{2}$, whereas for the second type, the variance is $4 \sigma^{2}$. Let $X_{1}, \ldots, X_{m}$ be $m$ independent growth observations on the first type [so $E\left(X_{i}\right)=\mu, V\left(X_{i}\right)=\sigma^{2}$ ], and let $Y_{1}, \ldots, Y_{n}$ be $n$ independent growth observations on the second type $\left[E\left(Y_{i}\right)=\mu, V\left(Y_{i}\right)=4 \sigma^{2}\right]$. Let $c$ be a numerical constant and consider the estimator $\hat{\mu}=c X+(1-c) Y$. For any $c$ between 0 and 1 this is a weighted average of the two sample means, e.g., $.7 X+.3 Y$
a. Show that for any $c$ the estimator is unbiased.
b. For fixed $m$ and $n$, what value $c$ minimizes $V(\hat{\mu})$ ? [Hint: The estimator is a linear combination of the two sample means and these means are independent. Once you have an expression for the variance, differentiate with respect to $c$.]

Clarissa Noh
Clarissa Noh
Numerade Educator
06:32

Problem 17

In Chapter 3 , we defined a negative binomial rv as the number of failures that occur before the $r$ th success in a sequence of independent and identical success/failure trials. The probability mass function (pmf) of $X$ is
a. Suppose that $r \geq 2$. Show that
$$
\hat{p}=(r-1) /(X+r-1)
$$
is an unbiased estimator for $p$. [Hint: Write out $E(\hat{p})$ and cancel $x+r-1$ inside the sum.]
b. A reporter wishing to interview five individuals who support a certain candidate begins asking people whether $(S)$ or not $(F)$ they support the candidate. If the sequence of responses is SFFSFFFSSS, estimate $p=$ the true proportion who support the candidate.

Robin Corrigan
Robin Corrigan
Numerade Educator
01:18

Problem 18

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a pdf $f(x)$ that is symmetric about $\mu$, so that $\widetilde{X}$ is an unbiased estimator of $\mu$. If $n$ is large, it can be shown that $V(\widetilde{X}) \approx 1 /\left\{4 n[f(\mu)]^{2}\right\}$. When the underlying pdf is Cauchy (see Example 7.8), $V(X)=\infty$, so $X$ is a terrible estimator. What is $V(\tilde{X})$ in this case when $n$ is large?

Manik Pulyani
Manik Pulyani
Numerade Educator
04:35

Problem 19

An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code. Having obtained a random sample of $n$ students, she realizes that asking each, "Have you violated the honor code?" will probably result in some untruthful responses. Consider the following scheme, called a randomized response technique. The investigator makes up a deck of 100 cards, of which 50 are of type I and 50 are of type II.
Type I: Have you violated the honor code (yes or no)?
Type II: Is the last digit of your telephone number a 0,1 , or 2 (yes or no)?
Each student in the random sample is asked to mix the deck, draw a card, and answer the resulting question truthfully. Because of the irrelevant question on type II cards, a yes response no longer stigmatizes the respondent, so we assume that responses are truthful. Let $p$ denote the proportion of honor-code violators (i.e., the probability of a randomly selected student being a violator), and let $\lambda=P$ (yes response). Then $\lambda$ and $p$ are related by $\lambda=.5 p+(.5)(.3)$.
a. Let $Y$ denote the number of yes responses, so $Y \sim \operatorname{Bin}(n, \lambda)$. Thus $Y / n$ is an unbiased estimator of $\lambda$. Derive an estimator for $p$ based on $Y$. If $n=80$ and $y=20$, what is your estimate? [Hint: Solve $\lambda=.5 p+.15$ for $p$ and then substitute $Y / n$ for $\lambda .$ ]
b. Use the fact that $E(Y / n)=\lambda$ to show that your estimator $\hat{p}$ is unbiased.
c. If there were 70 type I and 30 type II cards, what would be your estimator for $p$ ?

Robin Corrigan
Robin Corrigan
Numerade Educator
20:06

Problem 20

Return to the problem of estimating the population proportion $p$ and consider another adjusted estimator, namely
$$
\hat{p}=\frac{X+\sqrt{n / 4}}{n+\sqrt{n}}
$$
The justification for this estimator comes from the Bayesian approach to point estimation to be introduced in Section 14.4.
a. Determine the mean squared error of this estimator. What do you find interesting about this MSE?
b. Compare the MSE of this estimator to the MSE of the usual estimator (the sample proportion).

js
John Smith
Numerade Educator