Deck 5: Sampling Distributions

Question 1

The probability distribution shown below describes a population of measurements.
$\begin{array}{c|ccc}\mathrm{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\mathrm{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$
Suppose that we took repeated random samples of $n = 2$ observations from the population described above. Find the expected value of the sampling distribution of the sample mean.

A) 2
B) 0
C) 1
D) 3
E) 4

Answer 1

2

Question 2

The sampling distribution of the sample mean is shown below. Find the expected value of the sampling distribution of the sample mean.

A) 5
B) 4
C) 6
D) 7

Answer 2

C

Question 3

Which of the following describes what the property of unbiasedness means?

A) The shape of the sampling distribution is approximately normally distributed.
B) The sampling distribution in question has the smallest variation of all possible sampling distributions.
C) The center of the sampling distribution is found at the population standard deviation.
D) The center of the sampling distribution is found at the population parameter that is being estimated.

Answer 3

D

Question 4

Consider the population described by the probability distribution below. $\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}$

a. Find $\mu$ .
b. Find the sampling distribution of the sample mean $\bar { x }$ for a random sample of $n = 2$ measurements from the distribution.
c. Show that $\bar { x }$ is an unbiased estimator of $\mu$ .

Question 5

The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 10 minutes and a standard deviation of 1.6 minutes. If times were collected from 40 randomly selected walks, describe the sampling distribution of $\bar { x } ,$ the sample mean time.

Question 6

The probability distribution shown below describes a population of measurements.
$\begin{array}{c|ccc}\mathrm{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\mathrm{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$
Suppose that we took repeated random samples of $n = 2$ observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

A)
$\begin{array}{c|ccccc}\overline{\mathrm{x}} & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm{p}(\overline{\mathrm{x}}) & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5\end{array}$

B)
$\begin{array}{c|ccccc}\overline{\mathrm{x}} & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm{p}(\overline{\mathrm{x}}) & 2 / 9 & 2 / 9 & 1 / 9 & 2 / 9 & 2 / 9\end{array}$

C)
$\begin{array}{c|ccc}\bar{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\bar{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}$

Question 7

A random sample of size n is to be drawn from a population with $\mu = 700 \text { and } \sigma = 200$ What size sample would be necessary in order to reduce the standard error to 10?

Question 8

Consider the probability distribution shown here.

$\begin{array}{l|rrr}\hline x & 0 & 2 & 4 \\\hline p(x) & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\\hline\end{array}$

Let $\bar { x }$ be the sample mean for random samples of $n = 2$ measurements from this distribution. Find $E ( x )$ and $E ( \bar { x } )$ .

Question 9

Consider the population described by the probability distribution below. $$\begin{array} { c | c | c | c } x & 0 & 2 & 4 \\\hline p ( x ) & \frac { 1 } { 3 } & \frac { 1 } { 3 } & \frac { 1 } { 3 }\end{array}$$ a. Find $\mu$ .
b. Find the sampling distribution of the sample mean for a random sample of $n = 3$ measurements from this distribution.
c. Find the sampling distribution of the sample median for a random sample of $n = 3$ observations from this population.
d. Show that both the mean and the median are unbiased estimators of $\mu$ for this population.
e. Find the variances of the sampling distributions of the sample mean and the sample median.
f. Which estimator would you use to estimate $\mu$ ? Why?

Question 10

Suppose a random sample of $n = 64$ measurements is selected from a population with mean $\mu = 65$ and standard deviation $\sigma = 12$ . Find the $z$ -score corresponding to a value of $\bar { x }$ $= 68$ .

Question 11

The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean weight of these 100 bags exceeds 10.45 ounces.

Question 12

Which of the following describes what the property of minimum variance means?

A) The sampling distribution in question has the smallest variation of all possible unbiased sampling distributions.
B) The center of the sampling distribution is found at the population standard deviation.
C) The shape of the sampling distribution is approximately normally distributed.
D) The center of the sampling distribution is found at the population parameter that is being estimated.

Question 13

Consider the population described by the probability distribution below.

$\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}$

a. Find $\sigma ^ { 2 }$ .
b. Find the sampling distribution of the sample variance $s ^ { 2 }$ for a random sample of $n = 2$ measurements from the distribution.
c. Show that $s ^ { 2 }$ is an unbiased estimator of $\sigma ^ { 2 }$ .

Question 14

The probability distribution shown below describes a population of measurements that can assume values of 5, 9, 13, and 17, each of which occurs with the same frequency:

$\begin{array}{c|cccc}\hline x & 5 & 9 & 13 & 17 \\\hline p(x) & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\\hline\end{array}$

Find $E ( x ) = \mu$ . Then consider taking samples of $n = 2$ measurements and calculating $\bar { x }$ for each sample. Find the expected value, $E ( \bar { x } )$ , of $\bar { x }$ .

Question 15

Question 16

The probability distribution shown below describes a population of measurements that can assume values of 2, 5, 8, and 11, each of which occurs with the same frequency:

$\begin{array}{l|llll}\hline x & 2 & 5 & 8 & 11 \\\hline p(x) & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\\hline\end{array}$

Consider taking samples of $n = 2$ measurements and calculating $\bar { x }$ for each sample. Construct the probability histogram for the sampling distribution of $\bar { x }$ .

Question 17

Suppose students' ages follow a skewed right distribution with a mean of 23 years old and a standard deviation of 4 years. If we randomly sample 200 students, which of the following
Statements about the sampling distribution of the sample mean age is incorrect?

A) The standard deviation of the sampling distribution is equal to 4 years.
B) The mean of the sampling distribution is approximately 23 years old.
C) The shape of the sampling distribution is approximately normal.
D) All of the above statements are correct.

Question 18

Question 19

Consider the population described by the probability distribution below.

$\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}$

The random variable $x$ is observed twice. The observations are independent. The different samples of size 2 and their probabilities are shown below.

$\begin{array}{c}\begin{array}{cc}\hline \text { Sample } &\text { Probability } \\\hline 2,2 & .04 \\2,5& .10 \\2,7 & .06\\\hline \end{array}\begin{array}{cc}\hline \text { Sample } & \text { Probability } \\\hline 5,2 & .10 \\5,5 & .25 \\5,7 & .15 \\\hline \end{array}\begin{array}{cc}\hline \text { Sample } & \text { Probability } \\\hline 7,2 & .06 \\7,5 & .15 \\7,7 & .09 \\\hline \end{array}\end{array}$



Find the sampling distribution of the sample mean $\bar { x }$ .

Question 20

The length of time a traffic signal stays green (nicknamed the "green time") at a particular intersection follows a normal probability distribution with a mean of 200 seconds and the standard
Deviation of 10 seconds. Use this information to answer the following questions. Which of the
Following describes the derivation of the sampling distribution of the sample mean?

A) The means of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted.
B) The standard deviations of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted.
C) A single sample of sufficiently large size is randomly selected from the population of "green times" and its probability is determined.
D) The mean and median of a large randomly selected sample of "green times" are calculated. Depending on whether or not the population of "green times" is normally distributed, either
The mean or the median is chosen as the best measurement of center.

Question 21

The ideal estimator has the greatest variance among all unbiased estimators.

Question 22

Question 23

If $\bar{x}$ is a good estimator for µ, then we expect the values of x to cluster around µ.

Question 24

One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.7 million. Suppose a sample of 100 major league players was taken. Find the approximate probability that the average salary of the 100 players that year exceeded $1.1 million.

A) approximately 1
B) .2357
C) .7357
D) approximately 0

Question 25

The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, µ and ?, are statistics.

Question 26

The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently
large.

Question 27

When estimating the population mean, the sample mean is always a better estimate than the sample median.

Question 28

Which of the following does the Central Limit Theorem allow us to disregard when working with the sampling distribution of the sample mean?

A) The mean of the population distribution.
B) The shape of the population distribution.
C) The standard deviation of the population distribution.
D) All of the above can be disregarded when the Central Limit Theorem is used.

Question 29

The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the Central Limit Theorem to be used?

A) The population from which we are sampling must be normally distributed.
B) The population from which we are sampling must not be normally distributed.
C) The population size must be large .
D) The sample size must be large .

Question 30

The sample mean, $\bar{x}$ , is a statistic.

Question 31

The average score of all golfers for a particular course has a mean of 66 and a standard deviation of 3.5. Suppose 49 golfers played the course today. Find the probability that the average score of the 49 golfers exceeded 67.

A) .0228
B) .3707
C) .1293
D) .4772

Question 32

Which of the following statements about the sampling distribution of the sample mean is incorrect?

A) The sampling distribution is approximately normal whenever the sample size is sufficiently large (n ≥ 30).
B) The standard deviation of the sampling distribution is σ.
C) The mean of the sampling distribution is µ.
D) The sampling distribution is generated by repeatedly taking samples of size n and computing the sample means.

Question 33

The Central Limit Theorem is considered powerful in statistics because __________.

A) it works for any population distribution provided the sample size is sufficiently large
B) it works for any sample provided the population distribution is known
C) it works for any population distribution provided the population mean is known
D) it works for any sample size provided the population is normal

Question 34

As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well.

Question 35

The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.

Question 36

Consider the population described by the probability distribution below.
$\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}$

a. Find $\mu$ .
b. Find the sampling distribution of the sample median for a random sample of $n = 2$ observations from
this population.
c. Show that the median is an unbiased estimator of $\mu$ .

Question 37

A) approximately normal; 0.32, 0.0005
B) approximately normal; 0.32, 0.023
C) skewed right; 128, 9.33
D) skewed right; 0.32, 0.023

Question 38

In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.

Question 39

A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended
to estimate.

Question 40

A point estimator of a population parameter is a rule or formula which tells us how to use sample
data to calculate a single number that can be used as an estimate of the population parameter.

Question 41

A) .92; .011
B) .92; .003
C) .08; .011
D) .08; .003

Question 42

Sample statistics are random variables, because different samples can lead to different values of the sample statistics.

Question 43

The weight of corn chips dispensed into a 14-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 14.5 ounces and a standard deviation of 0.2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean weight of these 100 bags exceeds 14.6 ounces.

A) .3085
B) .6915
C) .1915
D) approximately 0

Question 44

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 1.7 cars and a standard deviation of 5. The number of cars running the red light was observed on 100 randomly chosen days and the mean number of cars calculated. Describe the
Sampling distribution of the sample mean.

A) approximately normal with mean $= 1.7$ and standard deviation $= 0.5$
B) shape unknown with mean $= 1.7$ and standard deviation $= 0.5$
C) shape unknown with mean $= 1.7$ and standard deviation $= 5$
D) approximately normal with mean $= 1.7$ and standard deviation $= 5$

Question 45

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $1500 and the standard deviation is $500. The distribution is skewed to the right due to several high volume days (football game days). Suppose that 100 days are randomly selected and the average daily revenue computed. Which of the following describes the sampling distribution of the sample mean?

A) normally distributed with a mean of $1500 and a standard deviation of $500
B) skewed to the right with a mean of $1500 and a standard deviation of $500
C) normally distributed with a mean of $150 and a standard deviation of $50
D) normally distributed with a mean of $1500 and a standard deviation of $50

Question 46

The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.

Question 47

The standard error of the sampling distribution of the sample mean is equal to ?, the standard deviation of the population.

Question 48

A random sample of n = 300 measurements is drawn from a binomial population with probability of success .43. Give the mean and the standard deviation of the sampling distribution of the sample Proportion, $\hat { \mathrm { p } }$

A) .43; .014
B) 57; .014
C) .43; .029
D) .57; .029

Question 49

A) .26; .025
B) .26; .011
C) .74; .011
D) .74; .025

Question 50

A random sample of n = 400 measurements is drawn from a binomial population with probability of success .21. Give the mean and the standard deviation of the sampling distribution of the sample Proportion, $\hat { \mathrm { p } }$

A) .79; .008
B) .21; .008
C) .79; .02
D) .21; .02

Question 51

The Central Limit Theorem is important in statistics because _____.

A) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the population
B) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size
C) for a large n, it says the population is approximately normal
D) for any size sample, it says the sampling distribution of the sample mean is approximately normal

Question 52

The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators.