Deck 5: Sampling Distributions

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سؤال
The probability distribution shown below describes a population of measurements.
x024p(x)1/31/31/3\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=2n = 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
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سؤال
The sampling distribution of the sample mean is shown below. <strong>The sampling distribution of the sample mean is shown below.   Find the expected value of the sampling distribution of the sample mean.</strong> A) 5 B) 4 C) 6 D) 7 <div style=padding-top: 35px> Find the expected value of the sampling distribution of the sample mean.

A) 5
B) 4
C) 6
D) 7
سؤال
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.
سؤال
Consider the population described by the probability distribution below. x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find μ\mu .
b. Find the sampling distribution of the sample mean xˉ\bar { x } for a random sample of n=2n = 2 measurements from the distribution.
c. Show that xˉ\bar { x } is an unbiased estimator of μ\mu .
سؤال
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 xˉ,\bar { x } , the sample mean time.
سؤال
The probability distribution shown below describes a population of measurements.
x024p(x)1/31/31/3\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=2n = 2 observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

A)
x01234p(x)1/51/51/51/51/5\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)
x01234p(x)2/92/91/92/92/9\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)
xˉ024p(xˉ)1/31/31/3\begin{array}{c|ccc}\bar{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\bar{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}
سؤال
A random sample of size n is to be drawn from a population with μ=700 and σ=200\mu = 700 \text { and } \sigma = 200 What size sample would be necessary in order to reduce the standard error to 10?
سؤال
Consider the probability distribution shown here.

x024p(x)131313\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 xˉ\bar { x } be the sample mean for random samples of n=2n = 2 measurements from this distribution. Find E(x)E ( x ) and E(xˉ)E ( \bar { x } ) .
سؤال
Consider the population described by the probability distribution below. x024p(x)131313\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 μ\mu .
b. Find the sampling distribution of the sample mean for a random sample of n=3n = 3 measurements from this distribution.
c. Find the sampling distribution of the sample median for a random sample of n=3n = 3 observations from this population.
d. Show that both the mean and the median are unbiased estimators of μ\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 μ\mu ? Why?
 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?  <div style=padding-top: 35px>
سؤال
Suppose a random sample of n=64n = 64 measurements is selected from a population with mean μ=65\mu = 65 and standard deviation σ=12\sigma = 12 . Find the zz -score corresponding to a value of xˉ\bar { x } =68= 68 .
سؤال
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.
سؤال
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.
سؤال
Consider the population described by the probability distribution below.

x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find σ2\sigma ^ { 2 } .
b. Find the sampling distribution of the sample variance s2s ^ { 2 } for a random sample of n=2n = 2 measurements from the distribution.
c. Show that s2s ^ { 2 } is an unbiased estimator of σ2\sigma ^ { 2 } .
سؤال
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:

x591317p(x)14141414\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)=μE ( x ) = \mu . Then consider taking samples of n=2n = 2 measurements and calculating xˉ\bar { x } for each sample. Find the expected value, E(xˉ)E ( \bar { x } ) , of xˉ\bar { x } .
سؤال
 <div style=padding-top: 35px>
سؤال
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:

x25811p(x)14141414\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=2n = 2 measurements and calculating xˉ\bar { x } for each sample. Construct the probability histogram for the sampling distribution of xˉ\bar { x } .
سؤال
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.
سؤال
 <div style=padding-top: 35px>
سؤال
Consider the population described by the probability distribution below.

x257p(x).2.5.3\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}

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

 Sample  Probability 2,2.042,5.102,7.06 Sample  Probability 5,2.105,5.255,7.15 Sample  Probability 7,2.067,5.157,7.09\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 xˉ\bar { x } .
سؤال
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.
سؤال
The ideal estimator has the greatest variance among all unbiased estimators.
سؤال
 <div style=padding-top: 35px>
سؤال
If xˉ\bar{x} is a good estimator for µ, then we expect the values of x to cluster around µ.
سؤال
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
سؤال
The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, µ and ?, are statistics.
سؤال
The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently
large.
سؤال
When estimating the population mean, the sample mean is always a better estimate than the sample median.
سؤال
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.
سؤال
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 .
سؤال
The sample mean, xˉ\bar{x} , is a statistic.
سؤال
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
سؤال
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.
سؤال
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
سؤال
As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well.
سؤال
The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.
سؤال
Consider the population described by the probability distribution below.
x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find μ\mu .
b. Find the sampling distribution of the sample median for a random sample of n=2n = 2 observations from
this population.
c. Show that the median is an unbiased estimator of μ\mu .
سؤال
<strong> </strong> 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 <div style=padding-top: 35px>

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
سؤال
In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.
سؤال
A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended
to estimate.
سؤال
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.
سؤال
<strong> </strong> A) .92; .011 B) .92; .003 C) .08; .011 D) .08; .003 <div style=padding-top: 35px>

A) .92; .011
B) .92; .003
C) .08; .011
D) .08; .003
سؤال
Sample statistics are random variables, because different samples can lead to different values of the sample statistics.
سؤال
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
سؤال
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= 1.7 and standard deviation =0.5= 0.5
B) shape unknown with mean =1.7= 1.7 and standard deviation =0.5= 0.5
C) shape unknown with mean =1.7= 1.7 and standard deviation =5= 5
D) approximately normal with mean =1.7= 1.7 and standard deviation =5= 5
سؤال
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
سؤال
The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.
سؤال
The standard error of the sampling distribution of the sample mean is equal to ?, the standard deviation of the population.
سؤال
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, p^\hat { \mathrm { p } }

A) .43; .014
B) 57; .014
C) .43; .029
D) .57; .029
سؤال
<strong> </strong> A) .26; .025 B) .26; .011 C) .74; .011 D) .74; .025 <div style=padding-top: 35px>

A) .26; .025
B) .26; .011
C) .74; .011
D) .74; .025
سؤال
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, p^\hat { \mathrm { p } }

A) .79; .008
B) .21; .008
C) .79; .02
D) .21; .02
سؤال
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
سؤال
The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators.
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Deck 5: Sampling Distributions
1
The probability distribution shown below describes a population of measurements.
x024p(x)1/31/31/3\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=2n = 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
2
2
The sampling distribution of the sample mean is shown below. <strong>The sampling distribution of the sample mean is shown below.   Find the expected value of the sampling distribution of the sample mean.</strong> A) 5 B) 4 C) 6 D) 7 Find the expected value of the sampling distribution of the sample mean.

A) 5
B) 4
C) 6
D) 7
C
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.
D
4
Consider the population described by the probability distribution below. x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find μ\mu .
b. Find the sampling distribution of the sample mean xˉ\bar { x } for a random sample of n=2n = 2 measurements from the distribution.
c. Show that xˉ\bar { x } is an unbiased estimator of μ\mu .
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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 xˉ,\bar { x } , the sample mean time.
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6
The probability distribution shown below describes a population of measurements.
x024p(x)1/31/31/3\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=2n = 2 observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

A)
x01234p(x)1/51/51/51/51/5\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)
x01234p(x)2/92/91/92/92/9\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)
xˉ024p(xˉ)1/31/31/3\begin{array}{c|ccc}\bar{x} & 0 & 2 & 4 \\\hline \mathrm{p}(\bar{x}) & 1 / 3 & 1 / 3 & 1 / 3\end{array}
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7
A random sample of size n is to be drawn from a population with μ=700 and σ=200\mu = 700 \text { and } \sigma = 200 What size sample would be necessary in order to reduce the standard error to 10?
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8
Consider the probability distribution shown here.

x024p(x)131313\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 xˉ\bar { x } be the sample mean for random samples of n=2n = 2 measurements from this distribution. Find E(x)E ( x ) and E(xˉ)E ( \bar { x } ) .
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9
Consider the population described by the probability distribution below. x024p(x)131313\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 μ\mu .
b. Find the sampling distribution of the sample mean for a random sample of n=3n = 3 measurements from this distribution.
c. Find the sampling distribution of the sample median for a random sample of n=3n = 3 observations from this population.
d. Show that both the mean and the median are unbiased estimators of μ\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 μ\mu ? Why?
 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?
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10
Suppose a random sample of n=64n = 64 measurements is selected from a population with mean μ=65\mu = 65 and standard deviation σ=12\sigma = 12 . Find the zz -score corresponding to a value of xˉ\bar { x } =68= 68 .
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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.
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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.
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13
Consider the population described by the probability distribution below.

x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find σ2\sigma ^ { 2 } .
b. Find the sampling distribution of the sample variance s2s ^ { 2 } for a random sample of n=2n = 2 measurements from the distribution.
c. Show that s2s ^ { 2 } is an unbiased estimator of σ2\sigma ^ { 2 } .
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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:

x591317p(x)14141414\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)=μE ( x ) = \mu . Then consider taking samples of n=2n = 2 measurements and calculating xˉ\bar { x } for each sample. Find the expected value, E(xˉ)E ( \bar { x } ) , of xˉ\bar { x } .
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15
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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:

x25811p(x)14141414\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=2n = 2 measurements and calculating xˉ\bar { x } for each sample. Construct the probability histogram for the sampling distribution of xˉ\bar { x } .
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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.
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18
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19
Consider the population described by the probability distribution below.

x257p(x).2.5.3\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}

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

 Sample  Probability 2,2.042,5.102,7.06 Sample  Probability 5,2.105,5.255,7.15 Sample  Probability 7,2.067,5.157,7.09\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 xˉ\bar { x } .
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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.
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21
The ideal estimator has the greatest variance among all unbiased estimators.
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22
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23
If xˉ\bar{x} is a good estimator for µ, then we expect the values of x to cluster around µ.
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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
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25
The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, µ and ?, are statistics.
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26
The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently
large.
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27
When estimating the population mean, the sample mean is always a better estimate than the sample median.
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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.
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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 .
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30
The sample mean, xˉ\bar{x} , is a statistic.
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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
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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.
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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
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34
As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well.
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35
The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.
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36
Consider the population described by the probability distribution below.
x357p(x).1.7.2\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}

a. Find μ\mu .
b. Find the sampling distribution of the sample median for a random sample of n=2n = 2 observations from
this population.
c. Show that the median is an unbiased estimator of μ\mu .
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37
<strong> </strong> 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

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
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38
In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.
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39
A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended
to estimate.
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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.
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41
<strong> </strong> A) .92; .011 B) .92; .003 C) .08; .011 D) .08; .003

A) .92; .011
B) .92; .003
C) .08; .011
D) .08; .003
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42
Sample statistics are random variables, because different samples can lead to different values of the sample statistics.
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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
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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= 1.7 and standard deviation =0.5= 0.5
B) shape unknown with mean =1.7= 1.7 and standard deviation =0.5= 0.5
C) shape unknown with mean =1.7= 1.7 and standard deviation =5= 5
D) approximately normal with mean =1.7= 1.7 and standard deviation =5= 5
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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
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46
The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.
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47
The standard error of the sampling distribution of the sample mean is equal to ?, the standard deviation of the population.
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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, p^\hat { \mathrm { p } }

A) .43; .014
B) 57; .014
C) .43; .029
D) .57; .029
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49
<strong> </strong> A) .26; .025 B) .26; .011 C) .74; .011 D) .74; .025

A) .26; .025
B) .26; .011
C) .74; .011
D) .74; .025
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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, p^\hat { \mathrm { p } }

A) .79; .008
B) .21; .008
C) .79; .02
D) .21; .02
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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
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52
The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators.
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