Question 1

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

Accepted Answer

The standard error of the sampling distribution of the sample mean is equal to the standard deviation of the population (σ) divided by the square root of the sample size (n), not equal to σ itself.

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

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 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

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution. This is why option A is correct.

Question 4

The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 15 minutes and a standard deviation of $1.4$ minutes. If times were collected from 60 randomly selected walks, describe the sampling distribution of $\bar { x }$, the sample mean time.

Accepted Answer

The answer of The amount of time it takes a...

Question 5

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 $\sigma ^ { 2 }$.

Accepted Answer

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

Question 6

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $3450 and the standard deviation is $300. 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 $345 and a standard deviation of $30
B)normally distributed with a mean of $3450 and a standard deviation of $30
C)skewed to the right with a mean of $3450 and a standard deviation of $300
D)normally distributed with a mean of $3450 and a standard deviation of $300

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (in this case, 100 days is considered large). The mean of the sampling distribution will be the same as the population mean ($3450), and the standard deviation of the sampling distribution (standard error) is the population standard deviation divided by the square root of the sample size, which is $300/√100 = $30.

Question 7

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

Accepted Answer

The ideal estimator, often referred to as the Best Linear Unbiased Estimator (BLUE), achieves the lowest variance among all unbiased estimators, according to the Gauss-Markov theorem.

Question 8

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

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size n becomes large, regardless of the shape of the population distribution. It does not guarantee that the population itself is normal.

Question 9

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 increases, the standard error of the sampling distribution of the sample mean decreases because the standard error is inversely proportional to the square root of the sample size.

Question 10

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

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

Question 11

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 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.

Accepted Answer

The Central Limit Theorem (CLT) allows us to disregard the shape of the population distribution when working with the sampling distribution of the sample mean, as it states that the sampling distribution of the sample mean will be approximately normally distributed regardless of the population distribution's shape, given a sufficiently large sample size.

Question 12

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)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.

Accepted Answer

The standard deviation of the sampling distribution, also known as the standard error, is calculated as the population standard deviation divided by the square root of the sample size. Given a population standard deviation of 3 years and a sample size of 350, the standard deviation of the sampling distribution would be less than 3 years, making statement C incorrect.

Question 13

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.
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

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution, as long as the sample size is sufficiently large (which is generally considered to be the case with n ≥ 30). Here, with a sample size of 100, the sampling distribution of the sample mean will be approximately normal. The mean of the sampling distribution (μx̄) is equal to the population mean (μ), which is 2.4. The standard deviation of the sampling distribution (σx̄), also known as the standard error, is equal to the population standard deviation (σ) divided by the square root of the sample size (n), which is 4/√100 = 0.4.

Question 14

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 answer of A random sample of size $n$ is...

Question 15

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

Accepted Answer

The minimum-variance unbiased estimator (MVUE) is defined as the estimator that has the smallest variance among all unbiased estimators for a parameter, ensuring it provides the most precise estimates without systematic error.

Question 16

The Central Limit Theorem is considered powerful in statistics because __________.
A)it works for any sample size provided the population is normal
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 population distribution provided the sample size is sufficiently large

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large. This makes it a powerful tool in statistics for making inferences about population parameters.

Question 17

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

Accepted Answer

The Central Limit Theorem applies regardless of the population's distribution shape, provided the sample size is sufficiently large (usually n ≥ 30 is considered large enough). This means that the sample mean will be approximately normally distributed if the sample size is large, even if the population itself is not normally distributed.

Question 18

Which of the following describes what the property of minimum variance means?
A)The center of the sampling distribution is found at the population standard deviation.
B)The shape of the sampling distribution is approximately normally distributed.
C)The center of the sampling distribution is found at the population parameter that is being estimated.
D)The sampling distribution in question has the smallest variation of all possible unbiased sampling distributions.

Accepted Answer

The property of minimum variance means that among all unbiased estimators, the estimator with the minimum variance is preferred because it has the least spread or variability around the parameter being estimated. This makes it the most precise or reliable estimator.

Question 19

Which of the following statements about the sampling distribution of the sample mean is incorrect? 
A) The mean of the sampling distribution is $\mu$.
B) The sampling distribution is generated by repeatedly taking samples of size $n$ and computing the sample means.
C) The standard deviation of the sampling distribution is $\sigma$.
D) The sampling distribution is approximately normal whenever the sample size is sufficiently large $( n \geq 30 )$.

Accepted Answer

The standard deviation of the sampling distribution, also known as the standard error, is $\sigma / \sqrt{n}$, not $\sigma$. This reflects how the variability of the sample mean decreases as the sample size increases.

Question 20

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...

Quiz 6: Sampling Distributions

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

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 $\bar { x }$ .

The Central Limit Theorem is important in statistics because _____.

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

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

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

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

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 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 ?

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

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

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?

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

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

Quiz 6: Sampling Distributions

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

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 .

The Central Limit Theorem is important in statistics because _____.

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

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

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

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

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 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 ?

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

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

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?

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

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

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 $\bar { x }$ .

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 ?