Question 1

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 2

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

Accepted Answer

The ideal estimator, also known as the efficient estimator, has the lowest variance among all unbiased estimators. This is a consequence of the Cramér-Rao inequality which states that the variance of any unbiased estimator is bounded from below by the inverse of the Fisher information. The ideal estimator achieves this lower bound, making it the estimator with the lowest variance.

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 μ. 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 μ 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? Why? 6.3 The Sampling Distribution of x-bar and the Central Limit Theorem 1 Understand Central Limit Theorem

Accepted Answer

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

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 14 minutes and a standard deviation of 1.1 minutes. If times were collected from 40 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

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 4.2 cars and a standard deviation of 6. 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 = 4.2 and standard deviation = 0.6
B) approximately normal with mean = 4.2 and standard deviation = 6
C) shape unknown with mean = 4.2 and standard deviation = 6
D) shape unknown with mean = 4.2 and standard deviation = 0.6

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 (n=100 is large), with the same mean as the population mean (4.2) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (6/√100 = 0.6).

Question 6

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

Accepted Answer

This is the definition of the MVUE. Its variance is guaranteed to be lower than any other unbiased estimator for the same parameter.

Question 7

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

Accepted Answer

The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases, but it does not guarantee that the population itself is normal.

Question 8

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $2700 and the standard deviation is $400. 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 $2700 and a standard deviation of $40
B) normally distributed with a mean of $2700 and a standard deviation of $400
C) normally distributed with a mean of $270 and a standard deviation of $40
D) skewed to the right with a mean of $2700 and a standard deviation of $400

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. The mean of the sampling distribution will be equal to the mean of the population, and the standard deviation of the sampling distribution will be equal to the standard deviation of the population divided by the square root of the sample size. In this case, the mean of the population is $2700 and the standard deviation of the population is $400. The sample size is 100, so the standard deviation of the sampling distribution will be $400 / sqrt(100) = $40. Therefore, the sampling distribution of the sample mean will be normally distributed with a mean of $2700 and a standard deviation of $40.

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

Accepted Answer

The Central Limit Theorem requires a large sample size for the sampling distribution of the sample mean to be approximately normal, regardless of the population's distribution.

Question 10

A random sample of size n is to be drawn from a population with μ = 1500 and σ = 200. What size sample would be necessary in order to reduce the standard error to 20? 2 Find Mean, Standard Deviation

Accepted Answer

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

Question 11

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

Accepted Answer

Actually, the standard error of the sampling distribution of the sample mean gets smaller as the sample size gets larger. This is because a larger sample size leads to a more accurate estimate of the true population mean.

Question 12

Suppose students' ages follow a skewed right distribution with a mean of 21 years old and a standard deviation of 2 years. If we randomly sample 450 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 2 years.
B) The shape of the sampling distribution is approximately normal.
C) The mean of the sampling distribution is approximately 21 years old.
D) All of the above statements are correct.

Accepted Answer

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 2/√450, not 2 years.

Question 13

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 shape of the population distribution.
B) The mean 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 14

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

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 15

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

Accepted Answer

The Central Limit Theorem states that for a large sample size (n), the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.

Question 16

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

Accepted Answer

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

Question 17

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 center of the sampling distribution is found at the population parameter that is being estimated.
D) The shape of the sampling distribution is approximately normally distributed.

Accepted Answer

Minimum variance means that the sampling distribution has the smallest possible variance among all unbiased estimators of a parameter.

Question 18

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 19

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

Accepted Answer

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

Question 20

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 σ/√n, where n is the sample size.

Quiz 6: Sampling Distributions

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

The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 14 minutes and a standard deviation of 1.1 minutes. If times were collected from 40 randomly selected walks, describe the sampling distribution of $\bar{x}$ , the sample mean time.

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

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

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 random sample of size n is to be drawn from a population with μ = 1500 and σ = 200. What size sample would be necessary in order to reduce the standard error to 20? 2 Find Mean, Standard Deviation

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

Suppose students' ages follow a skewed right distribution with a mean of 21 years old and a standard deviation of 2 years. If we randomly sample 450 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?

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 is considered powerful in statistics because __________.

The Central Limit Theorem is important in statistics because _____.

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

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?

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