Probability and Statistics with R

Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt

Chapter 6 Sampling and Sampling Distributions - all with Video Answers

Educators

Chapter Questions

01:12

Problem 1

How many ways can a host randomly choose 8 people out of 90 in the audience to participate in a TV game show?

Sanchit Jain

Numerade Educator

02:09

Problem 2

Let $X$ be a $t_5$.
(a) Find $\mathbb{P}(X<3)$.
(b) Calculate $\mathbb{P}(2<X<3)$.
(c) Find $a$ so that $\mathbb{P}(X<a)=0.05$.

K Joseph

Numerade Educator

01:08

Problem 3

If $(1-2 t)^{-5}, t<\frac{1}{2}$, is the mgf of a random variable $X$, find $\mathbb{P}(X<15.99)$.

Narayan Hari

Numerade Educator

04:59

Problem 4

If $X \sim \chi_{10}^2$, find the constants $a$ and $b$ so that $\mathbb{P}(a<X<b)=0.90$ and $\mathbb{P}(X<a)=0.05$.

Kira Schwander

Numerade Educator

03:34

Problem 5

Let $X$ be a $\chi_{10}^2$. Calculate $\mathbb{P}(X<8)$ and $\mathbb{P}(X>6)$. Calculate $a$ so that $\mathbb{P}(X<a)=.05$. What are the population mean and population variance of $X$ ?

Sarah X

Numerade Educator

09:00

Problem 6

Let $X$ be distributed as an $F_{2,5}$. Calculate $\mathbb{P}(X<1)$ and the median of $X$. Calculate $a$ so that $\mathbb{P}(X<a)=0.10$. What are the population mean and population variance of $X$ ?

Abhirup Pal

Numerade Educator

01:21

Assume a population with 5 elements:
$$
X_1=0, \quad X_2=1, \quad X_3=2, \quad X_4=3, \quad X_5=4 .
$$
(a) Calculate $\bar{X}$ and $\sigma^2$.
(b) Calculate the sampling distribution of the mean for random samples of size 3 taken without replacement. Verify that the mean of $\bar{X}$ is 2 and that the variance of $\bar{X}$ is $a^2 / 6$.
(c) Calculate the sampling distribution of $\bar{X}$ for random samples of size 3 taken with replacement. Verify that the mean of $\bar{X}$ is 2 and that the variance of $\bar{X}$ is $\sigma^2 / n$.

Akhil Choudhary

Numerade Educator

01:21

Problem 8

A population has the following elements: $2,5,8,12,13$.
(a) Enumerate all the samples of size 2 that can be drawn with and without replacement.
(b) Calculate the mean of the population.
(c) Calculate the variance of the population.
(d) Calculate the standard deviation of the population.
(e) Calculate the mean of the sample mean, $E[\bar{X}]$.
(f) Calculate the variance of the sampled mean, $\operatorname{Var}(\bar{X})$.
(g) Calculate the standard deviation of the sample mean.
(h) Calculate the mean of the sample variance, $E\left[S^2\right]$.
(i) Is the variance of $\bar{X}$ larger when sampling with or without replacement? Explain your answer.

Akhil Choudhary

Numerade Educator

01:22

Problem 9

Determine whether the following expressions are statistics or not:
(a) $\sum_{i=1}^n X_i$
(b) $\sum_{i=1}^n X_i-\bar{X}$
(c) $\bar{X}-\sigma$
(d) $X_1+X_2 / 6$

Manik Pulyani

Numerade Educator

03:21

Problem 10

Use the data frame wheatUSA2004 from the PASWR package; draw all samples of sizes 2, 3, and 4 ; and calculate the mean of the means. What size provides the best approximation to the population mean? What is the variance of these means?

Sheryl Ezze

Numerade Educator

01:22

Problem 11

Given a random sample of size 6 from $N(0, \sigma)$, calculate
(a) $\mathbb{P}\left(\frac{\bar{x}}{S}>2\right)$ and
(b) $\mathbb{P}\left(\left|\frac{\bar{X}}{S_u}\right| \leq 4\right)$.

Tyler Moulton

Numerade Educator

22:07

Problem 12

Constant velocity joints (CV joints) allow a rotating shaft to transmit power through a variable angle, at constant rotational speed, without an appreciable increase in friction or play. An after-market company produces CV joints. To optimize energy transfer, the drive shaft must be very precise. The company has two different branches that produce $\mathrm{CV}$ joints where the variability of the drive shaft is known to be $2 \mathrm{~mm}$. A sample of $n_1=10$ is drawn from the first branch, and a sample of $n_2=15$ is drawn from the second branch. Suppose that the diameter follows a normal distribution. What is the probability that the drive shafts coming from the first branch will have greater variability than those of the second branch?

Khoobchandra Agrawal

Numerade Educator

04:18

Problem 13

Given a population $N(\mu, \sigma)$ with unknown mean and variance, a sample of size 11 is drawn and the sample variance $S^2$ is calculated. Calculate the probability $\mathbb{P}(0.5<$ $\left.s^2 / \sigma^2<1.2\right)$.

Sheryl Ezze

Numerade Educator

00:55

Problem 14

Simulate 20,000 random samples of sizes $30,100,300$, and 500 from an expontial distribution with a mean of $1 / 5$. Estimate the density of the sampling distribution with the function density(). Superimpose a theoretical normal density with appropriate mean and standard deviation. What sample size is needed to get an estimated density close to a normal density?

Christopher Stanley

Numerade Educator

09:57

Problem 15

The plastic tubes produced by company $X$ for the irrigation system used in golf courses have a length of 1.5 meters and a standard deviation of 0.1 meter. The plastic tubes produced by company $Y$ have a length of 1 meter and a standard deviation of 0.09 meter. Suppose that both tube lengths follow normal distributions.
(a) Calculate the probability that a random sample of 15 tubes from company $X$ has a mean length at least 0.45 meter greater than the mean length of a random sample of size 20 from company $Y$.
(b) Suppose that the population variances are unknown but equal, $S_x=0.1$, and $S_y=0.09$. Calculate the probability that a random sample of 15 plastic tubes from company $X$ has a mean length at least 0.45 meter greater than the mean length of a random sample of 20 plastic tubes from company $Y$.

Chris Trentman

Numerade Educator

04:10

Problem 16

Plot the density function of an $F_{4,6}$ random variable. Find the area to the left of $x=3$ and shade this region in the original plot.

Anurag Kumar

Numerade Educator

03:04

Problem 17

Let $X_1, X_2, X_3, X_4$ be a random sample from a $N(0, \sigma)$. Calculate the distribution of $\frac{\left(X_1-X_2\right)^2}{\left(X_3+X_4\right)^2}$

Ameer Said

Numerade Educator

01:18

Problem 18

Let $X_1, X_2, X_3, X_4, X_5, X_6$ be a random sample drawn from a $N\left(0, \sigma^2\right)$ population. Find the values of $c$ so that the statistic $\frac{c X_1+X_2+X_1}{\sqrt{X_4^2+X_5^2+X_6^2}}$ follows a $t_3$-distribution.

Gregory Higby

Numerade Educator

02:58

Problem 19

Consider a random sample of size $n$ from an exponential distribution with parameter $\lambda$. Use moment generating functions to show that the sample mean follows a $\Gamma(n, \lambda n)$. Graph the theoretical sampling distribution of $\bar{X}$ when sampling from an $\operatorname{Exp}(\lambda=1)$ for $n=30,100,300$, and 500. Superimpose an appropriate normal density for each $\Gamma(n, \lambda n)$. At what sample size do the sampling distribution and superimposed density virtually coincide?

Idabelle Cunningham

Numerade Educator

03:14

Problem 20

Set the seed equal to 10 , and simulate 20,000 random samples of size $n_x=65$ from a $N\left(4, \sigma_x=\sqrt{2}\right), 20,000$ random samples of size $n_y=90$ from a $N\left(5, \sigma_y=\sqrt{3}\right)$ and verify that the simulated statistic $\frac{S_s^2 / \sigma_x^2}{S_y^2 / \sigma_y^2}$ follows an $F_{64,89}$ distribution.

Robin Corrigan

Numerade Educator

02:58

Problem 21

Set the seed equal to 95 , and simulate $m=20,000$ random samples of size $n=1000$ from a Bernoulli $(\pi=0.4)$. Verify that the sample proportion follows an approximate normal distribution with a mean approximately equal to 0.4 and a standard deviation approximately equal to 0.01549 .

Idabelle Cunningham

Numerade Educator

01:07

Problem 22

A communication system consists of $n$ components, where the probability that each component works is $\pi$. The system will work if at least half of its components work. For what values of $\pi$ will a system consisting of 5 components have a greater probability of working than a system consisting of 3 components? Plot the probability each system ( $n=5$ and $n=3$ ) works for values of $\pi$ from 0 to 1 in increments of 0.01 .

Amrita Bhasin

Numerade Educator

01:49

Problem 23

Given $X \sim N(0, \sigma=1), Y \sim N(2, \sigma=2)$, and $Z \sim N(4, \sigma=3)$, what is the distribution of $W=X+Y+Z$ ? Set the seed equal to 368 and simulate 1000 samples, each of size 1 for $X, Y$, and $Z$. Add the values in the three vectors to obtain $W$ 's empirical distribution. Create a density histogram of the simulated values of $W$ and superimpose the theoretical density of $W$.

Robin Corrigan

Numerade Educator

01:27

Problem 24

Set the seed equal to 48 , and simulate a $\chi_3^2$ distribution by summing the squares of three simulated standard normal random variables, each having length 20,000 . Create a density histogram of the simulated $\chi_3^2$ random variable. Superimpose the theoretical $\chi_3^2$ density over the histogram.

Angela Guo

Numerade Educator

02:32

Problem 25

Verify empirically that
$$
\frac{N(0,1)}{\left(\frac{1}{5} \chi_5^2\right)^{\frac{1}{2}}} \sim t_5
$$
by setting the seed equal to 36 and generating a sample of size 1000 from a $N(0,1)$ distribution. Generate another sample of size 1000 from a $\chi_5^2$ distribution. Perform the appropriate arithmetic to arrive at the simulated sampling distribution. Create a density histogram of the results and superimpose a theoretical $t_5$ density.

Jacob Fry

Numerade Educator

View

Problem 26

A farmer is interested in knowing the mean weight of his chickens when they leave the farm. Suppose that the standard deviation of the chickens' weight is 500 grams.
(a) What is the minimum number of chickens needed to ensure the a standard deviation of the mean is no more than 100 grams with a confidence level of 0.95 ?
(b) If the farm has three coops and the mean chicken weight in each coop is 1.8, 1.9, and $2 \mathrm{~kg}$, respectively, calculate the probability that a random sample of 50 chickens with an average weight larger than $1.975 \mathrm{~kg}$ comes from the first coop. Assume the weight of the chickens follows a normal distribution.

Rashmi Sinha

Numerade Educator

01:15

Problem 27

Find the required sample size $(n)$ to estimate the proportion of students spending more than $€ 10$ a week on entertainment with a $95 \%$ confidence interval so that the margin of error is no more than 0.02 .

Lucas Finney

Numerade Educator

01:08

Problem 28

$15.3 \%$ of the Spanish Internet domain names are ".org." If a sample of 2000 Spanish domain names is taken,
(a) Calculate the exact probability that at least 200 domain names will be ".org.".
(b) Compute an approximate answer that at least 200 domain names will be ".org." with a normal approximation.

Hoan Nguyen

Numerade Educator

02:33

Problem 29

Set the seed equal to 86 , and simulate $m_1=20,000$ samples of size $n_1=1000$ from a $\operatorname{Bin}\left(n_1, \pi=0.3\right)$ and $m_2=20,000$ samples of size $n_2=1100$ from a $\operatorname{Bin}\left(n_2, \pi=0.7\right)$. Verify that the difference of sampling proportions follows a normal distribution.

Jameson Kuper

Numerade Educator

04:35

Problem 30

Given a random sample of size $n$ from an exponential distribution with parameter $\lambda$, prove that the sample mean follows a $\Gamma(n, \lambda n)$. Set the seed equal to 679 , and simulate $m=1000$ random samples of size $n=100$ from an $\operatorname{Exp}(\lambda=1)$, and check that the normal approximation of the mean is appropriate. Repeat this exercise with random samples of size $n=3$, and verify that, in this case, $\Gamma(3,3)$ is more appropriate to use than the normal distribution.

Jameson Kuper

Numerade Educator

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