Question 1

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

Accepted Answer

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

Question 2

Suppose a random sample of $n = 36$ measurements is selected from a population with mean $\mu = 256$ and variance $\sigma ^ { 2 } = 144$. Find the mean and standard deviation of the sampling distribution of the sample mean $\bar { x }$.

Accepted Answer

The answer of Suppose a random sample of \(n =...

Question 3

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

Accepted Answer

If $\bar{x}$ is a good estimator for µ, it means that the sample mean is close to the population mean, implying that the sample values (x) are likely to be distributed around µ.

Question 4

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

Accepted Answer

The sample mean's distribution is normal with mean $1.6 million and standard deviation $0.07 million (0.7/√100). The z-score for $1.1 million is (1.1-1.6)/0.07 = -7.14, which is far in the left tail, making the probability of exceeding $1.1 million approximately 1.

Question 5

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

Accepted Answer

In a binomial experiment, the probability of success, p, is a parameter, but the mean (µ = np) and standard deviation (σ = √(np(1-p))) are also parameters, not statistics. Statistics are calculated from sample data, while parameters describe the entire population or distribution.

Question 6

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

Accepted Answer

The Central Limit Theorem guarantees that the sampling distribution of the mean will be approximately normal when the sample size is sufficiently large, not the population itself.

Question 7

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

Accepted Answer

The sample mean is sensitive to outliers and may not always provide a better estimate of the population mean than the sample median, especially in skewed distributions. The median can be a more robust estimator in such cases.

Question 8

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.

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution, given a sufficiently large sample size.

Question 9

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 .

Accepted Answer

The Central Limit Theorem applies when the sample size is large, typically n ≥ 30, regardless of the population's distribution.

Question 10

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

Accepted Answer

The sample mean, $\bar{x}$, is a statistic because it is a numerical value calculated from a sample, used to estimate the population mean.

Question 11

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

Accepted Answer

The standard error of the mean is 3.5/sqrt(49) = 0.5. The z-score for 67 is (67-66)/0.5 = 2. The probability of a z-score exceeding 2 is 0.0228.

Question 12

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.

Accepted Answer

The standard deviation of the sampling distribution is not σ; it is σ/√n, where σ is the population standard deviation and n is the sample size.

Question 13

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

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution, if the sample size is sufficiently large.

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

This statement is true. The sampling distribution of a sample statistic is the theoretical probability distribution of all possible values of the statistic that could be calculated from samples of the same size and taken from the same population.

Question 16

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 $\mu$.

Accepted Answer

The answer of Consider the population described by the probability...

Question 17

Suppose a random sample of $\mathrm { n }$ measurements is selected from a binomial population with probability of success $\mathrm { p } = .32$. Given $\mathrm { n } = 400$, describe the shape, and find the mean and the standard deviation of the sampling distribution of the sample proportion, $\hat { p }$.
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

Accepted Answer

The answer of Suppose a random sample of \(\mathrm {...

Question 18

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

Accepted Answer

In most cases, the true mean and standard deviation of a population are not known and have to be estimated through various statistical techniques.

Question 19

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

Accepted Answer

A statistic is considered unbiased if the mean of its sampling distribution is equal to the parameter it is intended to estimate. Bias occurs when there is a systematic difference between the estimator's expected value and the true parameter value.

Question 20

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

This statement is true. A point estimator is a method of using sample data to calculate a single value that is used as the estimate of the population parameter.

Quiz 5: Sampling Distributions

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

Suppose a random sample of $n = 36$ measurements is selected from a population with mean $\mu = 256$ and variance $\sigma ^ { 2 } = 144$ . Find the mean and standard deviation of the sampling distribution of the sample mean $\bar { x }$ .

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

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.

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

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

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

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

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?

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

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.

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

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

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

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

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

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

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.

Quiz 5: Sampling Distributions

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

Suppose a random sample of n=36n = 36n=36 measurements is selected from a population with mean μ=256\mu = 256μ=256 and variance σ2=144\sigma ^ { 2 } = 144σ2=144 . Find the mean and standard deviation of the sampling distribution of the sample mean xˉ\bar { x }xˉ .

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

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.

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

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

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

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

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?

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

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.

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

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

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

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

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

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

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.

Suppose a random sample of $n = 36$ measurements is selected from a population with mean $\mu = 256$ and variance $\sigma ^ { 2 } = 144$ . Find the mean and standard deviation of the sampling distribution of the sample mean $\bar { x }$ .

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

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