Question 1

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

Accepted Answer

A) True 
 B)False  True

Question 2

The probability distribution shown below describes a population of measurements. $$\begin{array} { c | c c c } 
x & 0 & 2 & 4 \
\hline p ( x ) & 1 / 3 & 1 / 3 & 1 / 3
\end{array}$$ 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.

Accepted Answer

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

Question 3

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?

Accepted Answer

a. $\quad \mu = E ( x ) = \frac { 1 } { 3 } ( 0 ) + \frac { 1 } { 3 } ( 2 ) + \frac { 1 } { 3 } ( 4 ) = 2$
b.
$\begin{array}{c|c|c|c|c|c|c|c}
\bar{x} & 0 & \frac{2}{3} & \frac{4}{3} & 2 & \frac{8}{3} & \frac{10}{3} & 4 \
\hline p(\bar{x}) & \frac{1}{27} & \frac{1}{9} & \frac{2}{9} & \frac{7}{27} & \frac{2}{9} & \frac{1}{9} & \frac{1}{27}
\end{array}$
c.
$\begin{array}{c|c|c|c}
M & 0 & 2 & 4 \
\hline p(M) & \frac{7}{27} & \frac{13}{27} & \frac{7}{27}
\end{array}$

d. $E ( \bar { x } ) = \frac { 1 } { 27 } ( 0 ) + \frac { 1 } { 9 } \left( \frac { 2 } { 3 } \right) + \frac { 2 } { 9 } \left( \frac { 4 } { 3 } \right) + \frac { 7 } { 27 } ( 2 ) + \frac { 2 } { 9 } \left( \frac { 8 } { 3 } \right) + \frac { 1 } { 9 } \left( \frac { 10 } { 3 } \right) + \frac { 1 } { 27 } ( 4 ) = 2$; Since $E ( \bar { x } ) = \mu$, the sample
mean is an unbiased estimator of $\mu$.
$E ( M ) = \frac { 7 } { 27 } ( 0 ) + \frac { 13 } { 27 } ( 2 ) + \frac { 7 } { 27 } ( 4 ) = 2$; Since $E ( M ) = \mu$, the sample median is an unbiased estimator
of $\mu$.
e. $\quad \sigma \frac { 2 } { x } = \frac { 1 } { 27 } ( 0 - 2 ) ^ { 2 } + \frac { 1 } { 9 } \left( \frac { 2 } { 3 } - 2 \right) ^ { 2 } + \frac { 2 } { 9 } \left( \frac { 4 } { 3 } - 2 \right) ^ { 2 } + \frac { 7 } { 27 } ( 2 - 2 ) ^ { 2 } + \frac { 2 } { 9 } \left( \frac { 8 } { 3 } - 2 \right) ^ { 2 }$ $+ \frac { 1 } { 9 } \left( \frac { 10 } { 3 } - 2 \right) ^ { 2 } + \frac { 1 } { 27 } ( 4 - 2 ) ^ { 2 } = \frac { 8 } { 9 }$
$$\sigma _ { M } ^ { 2 } = \frac { 7 } { 27 } ( 0 - 2 ) ^ { 2 } + \frac { 13 } { 27 } ( 2 - 2 ) ^ { 2 } + \frac { 7 } { 27 } ( 4 - 2 ) ^ { 2 } = \frac { 56 } { 27 }$$
f. sample mean; The variance is smaller.

Question 4

The weight of corn chips dispensed into a 16-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 16.5 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 16.6 ounces.

Accepted Answer

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

Question 5

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.

Accepted Answer

A) True 
 B)False

Question 6

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 2.4 cars and a standard deviation of 4. 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.

Accepted Answer

A) approximately normal with mean = 2.4 and standard deviation = 4 
B) shape unknown with mean = 2.4 and standard deviation = 0.4 
C) approximately normal with mean = 2.4 and standard deviation = 0.4 
D) shape unknown with mean = 2.4 and standard deviation = 4 
A) approximately normal with mean = 2.4 and standard deviation = 4 
B) shape unknown with mean = 2.4 and standard deviation = 0.4 
C) approximately normal with mean = 2.4 and standard deviation = 0.4 
D) shape unknown with mean = 2.4 and standard deviation = 4

Question 7

The probability of success, $p$, in a binomial experiment is a parameter, while the mean and standard deviation, $\mu$ and $\sigma$, are statistics.

Accepted Answer

A) True 
 B)False

Question 8

Suppose studentsʹ ages follow a skewed right distribution with a mean of 24 years old and a standard deviation of 3 years. If we randomly sample 350 students, which of the following
Statements about the sampling distribution of the sample mean age is incorrect?

Accepted Answer

A) The mean of the sampling distribution is approximately 24 years old. 
B) The shape of the sampling distribution is approximately normal. 
C) The standard deviation of the sampling distribution is equal to 3 years. 
D) All of the above statements are correct. 
A) The mean of the sampling distribution is approximately 24 years old. 
B) The shape of the sampling distribution is approximately normal. 
C) The standard deviation of the sampling distribution is equal to 3 years. 
D) All of the above statements are correct.

Question 9

A random sample of $\mathrm { n } = 600$ measurements is drawn from a binomial population with probability of success .08. Give the mean and the standard deviation of the sampling distribution of the sample proportion, p.

Accepted Answer

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

Question 10

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

Accepted Answer

A) True 
 B)False

Question 11

The Central Limit Theorem is important in statistics because _____.

Accepted Answer

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

Question 12

The length of time a traffic signal stays green (nicknamed the ʺgreen timeʺ)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?

Accepted Answer

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

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

The sampling distribution of the sample mean is shown below. $$\begin{array} { c | c c c c c } 
\bar { x } & 4 & 5 & 6 & 7 & 8 \
\hline \mathrm { p } ( \bar { x } ) & 1 / 9 & 2 / 9 & 3 / 9 & 2 / 9 & 1 / 9
\end{array}$$ Find the expected value of the sampling distribution of the sample mean.

Accepted Answer

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

Question 15

A random sample of size $n$ is to be drawn from a population with $\mu = 1200$ and $\sigma = 200$. What size sample would be necessary in order to reduce the standard error to 25 ?

Accepted Answer

The standard error i

Question 16

The probability distribution shown below describes a population of measurements that
can assume values of 1, 5, 9, and 13, each of which occurs with the same frequency: $\begin{array}{l|cccc}
\hline x & 1 & 5 & 9 & 13 \
\hline p(x) & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \
\hline
\end{array}$

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

Accepted Answer

The answer of The probability distribution shown below describes a...

Question 17

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

Accepted Answer

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

Question 18

Suppose a random sample of $\mathrm { n }$ measurements is selected from a binomial population with probability of success $p = .35$. Given $n = 200$, describe the shape, and find the mean and the standard deviation of the sampling distribution of the sample proportion, $\hat { p }$.

Accepted Answer

A)  skewed right; $0.35,0.034$ 
B)  approximately normal; $0.35,0.034$ 
C)  approximately normal; $0.35,0.0011$ 
D)  skewed right; $70,6.75$ 
A)  skewed right; $0.35,0.034$ 
B)  approximately normal; $0.35,0.034$ 
C)  approximately normal; $0.35,0.0011$ 
D)  skewed right; $70,6.75$

Question 19

One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.8 million. Suppose a sample of 400 major league players was taken. Find the
Approximate probability that the average salary of the 400 players that year exceeded $1.1 million.

Accepted Answer

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

Question 20

The probability distribution shown below describes a population of measurements that
can assume values of 5, 10, 15, and 20, each of which occurs with the same frequency: $\begin{array}{l|cccc}
\hline x & 5 & 10 & 15 & 20 \
\hline p(x) & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \
\hline
\end{array}$

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 $\bar { x }$.

Accepted Answer

\[\begin{array} { l } 
E ( x ) = ( 5 ) \

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

The probability distribution shown below describes a population of measurements. x 0 2 4 p(x) 1/3 1/3 1/3 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 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.

The probability of success, $p$ , in a binomial experiment is a parameter, while the mean and standard deviation, $\mu$ and $\sigma$ , are statistics.

Suppose studentsʹ ages follow a skewed right distribution with a mean of 24 years old and a standard deviation of 3 years. If we randomly sample 350 students, which of the following Statements about the sampling distribution of the sample mean age is incorrect?

A random sample of $\mathrm { n } = 600$ measurements is drawn from a binomial population with probability of success .08. Give the mean and the standard deviation of the sampling distribution of the sample proportion, p.

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

The Central Limit Theorem is important in statistics because _____.

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

The sampling distribution of the sample mean is shown below. 4 5 6 7 8 () 1/9 2/9 3/9 2/9 1/9 Find the expected value of the sampling distribution of the sample mean.

A random sample of size $n$ is to be drawn from a population with $\mu = 1200$ and $\sigma = 200$ . What size sample would be necessary in order to reduce the standard error to 25 ?

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

Suppose a random sample of $\mathrm { n }$ measurements is selected from a binomial population with probability of success $p = .35$ . Given $n = 200$ , describe the shape, and find the mean and the standard deviation of the sampling distribution of the sample proportion, $\hat { p }$ .

One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.8 million. Suppose a sample of 400 major league players was taken. Find the Approximate probability that the average salary of the 400 players that year exceeded $1.1 million.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Multiple Regression and Model Building

Categorical Data Analysis

Nonparametric Statistics Available Online

Filters

Exam 6: Sampling Distributions

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

The probability distribution shown below describes a population of measurements. x 0 2 4 p(x) 1/3 1/3 1/3 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 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.

The probability of success, ppp , in a binomial experiment is a parameter, while the mean and standard deviation, μ\muμ and σ\sigmaσ , are statistics.

Suppose studentsʹ ages follow a skewed right distribution with a mean of 24 years old and a standard deviation of 3 years. If we randomly sample 350 students, which of the following Statements about the sampling distribution of the sample mean age is incorrect?

A random sample of n=600\mathrm { n } = 600n=600 measurements is drawn from a binomial population with probability of success .08. Give the mean and the standard deviation of the sampling distribution of the sample proportion, p.

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

The Central Limit Theorem is important in statistics because _____.

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

The sampling distribution of the sample mean is shown below. 4 5 6 7 8 () 1/9 2/9 3/9 2/9 1/9 Find the expected value of the sampling distribution of the sample mean.

A random sample of size nnn is to be drawn from a population with μ=1200\mu = 1200μ=1200 and σ=200\sigma = 200σ=200 . What size sample would be necessary in order to reduce the standard error to 25 ?

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

One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.8 million. Suppose a sample of 400 major league players was taken. Find the Approximate probability that the average salary of the 400 players that year exceeded $1.1 million.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Multiple Regression and Model Building

Categorical Data Analysis

Nonparametric Statistics Available Online

Filters

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

The probability of success, $p$ , in a binomial experiment is a parameter, while the mean and standard deviation, $\mu$ and $\sigma$ , are statistics.

A random sample of $\mathrm { n } = 600$ measurements is drawn from a binomial population with probability of success .08. Give the mean and the standard deviation of the sampling distribution of the sample proportion, p.

A random sample of size $n$ is to be drawn from a population with $\mu = 1200$ and $\sigma = 200$ . What size sample would be necessary in order to reduce the standard error to 25 ?