Question 1

A randomly selected value of $\bar { X }$ is likely to be ____________________ to(than)the mean of the population than is a randomly selected value of X.

Accepted Answer

The answer of A randomly selected value of \(\bar {...

Question 2

Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal.
A) It has the same shape,mean and standard deviation as the population.
B) It has the same mean as the population,but a different shape and standard deviation.
C) It the same mean and standard deviation as the population,but a different shape.
D) It has the same shape and mean as the population,but a different standard deviation.

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the mean for a large sample size (>30) will approach a normal distribution regardless of the shape of the population distribution. However, the mean of this sampling distribution will be equal to the mean of the population. 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. Therefore, the shape and standard deviation of the sampling distribution will be different from the population, but the mean will be the same. Choice B correctly reflects this.

Question 3

The width of the sampling distribution of $\bar { X }$ gets ____________________ as the sample size increases.

Accepted Answer

The answer of The width of the sampling distribution of...

Question 4

The variance of the sampling distribution of $\bar { X }$ is ____________________ the variance of the population we're sampling from for all sample sizes.

Accepted Answer

The answer of The variance of the sampling distribution of...

Question 5

Which of the following is true about the sampling distribution of the sample mean?
A) Its mean is always equal to
B) Its standard error is always equal to the population standard deviation $\sigma$.
C) Its shape is exactly normal if n is large enough.
D) None of these choices.

Accepted Answer

The answer of Which of the following is true about...

Question 6

The accuracy of the approximation of $\bar { X }$ with a normal distribution depends on the probability distribution of the ____________________ and on the sample ____________________.

Accepted Answer

The answer of The accuracy of the approximation of \(\bar...

Question 7

As n gets larger,the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.This phenomenon is due to the ____________________.

Accepted Answer

The answer of As n gets larger,the sampling distribution of...

Question 8

For sample sizes greater than 30,the sampling distribution of the mean is approximately normally distributed:
A) regardless of the shape of the population.
B) only if the shape of the population is symmetric.
C) only if the population is normally distributed.
D) None of these choices.

Accepted Answer

This is due to the Central Limit Theorem, which states that the sampling distribution of the mean will be approximately normally distributed regardless of the population distribution, given a sufficiently large sample size (usually considered to be greater than 30).

Question 9

The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.84 pounds.If a sample of 16 fish is taken,what is the standard error of the mean weight?
A) 0.840
B) 0.053
C) 0.210
D) None of these choices.

Accepted Answer

Standard error of the mean formula is: standard deviation/√sample size
Therefore, the standard error of the mean = 0.84/√16 = 0.21 (rounded to two decimal places). Thus, the correct answer is (C) 0.210.

Question 10

Because the value of the ____________________ varies randomly from sample to sample,we can regard it as a new random variable created by sampling.

Accepted Answer

The answer of Because the value of the ____________________ varies...

Question 11

The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.8 pounds.If a sample of 25 roasts yields a mean of 3.6 pounds,what is the Z-score for this sample mean?
A) -2.50
B) 2.50
C) -0.50
D) None of these choices.

Accepted Answer

The formula to calculate the Z-score for a sample mean is:

Z = (x̄ - μ) / (σ / sqrt(n))

where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the given values, we get:

Z = (3.6 - 3.2) / (0.8 / sqrt(25))
Z = 0.4 / 0.16
Z = 2.5

Therefore, the Z-score for this sample mean is 2.50, which corresponds to choice B.

Question 12

The standard error of the mean for a sample of 100 is 25.In order to cut the standard error of the mean in half (to 12.5)we must:
A) increase the sample size to 200.
B) decrease the sample size to 50.
C) keep the sample size at 100 and change something else.
D) None of these choices.

Accepted Answer

The answer of The standard error of the mean for...

Question 13

Suppose that 100 items are drawn from a population of manufactured products and the weight,X,of each item is recorded.Prior experience has shown that the weight has a non-normal probability distribution with $\mu$ = 8 ounces and $\sigma$ = 3 ounces.Which of the following is true about the sampling of $\bar { X }$ ?
A) Its mean is 8 ounces.
B) Its standard error is 0.3 ounces.
C) Its shape is approximately normal.
D) All of these choices are true.

Accepted Answer

A) The mean weight of the population is 8 ounces, so by the properties of the sample mean, the mean of the sampling distribution of the sample mean (i.e., the mean of $\bar{X}$) is also 8 ounces.

B) The standard error of the sample mean can be calculated using the formula $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. In this case, $\sigma$ is given as 3 ounces and the sample size is 100, so the standard error is $\frac{3}{\sqrt{100}} = 0.3$ ounces.

C) The central limit theorem states that, under certain conditions, the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution. One of these conditions is that the sample size is sufficiently large. In this case, the sample size is 100, which is generally considered to be large enough for the central limit theorem to apply.

Therefore, all three choices are true.

Question 14

As the number of throws of a fair die increases,the probability that the sample mean is close to (the number)____________________ increases.

Accepted Answer

The answer of As the number of throws of a...

Question 15

Suppose the ages of students in your program follow a positively skewed distribution with mean of 24 years and a standard deviation of 4 years.If we randomly sampled 100 students,which of the following statements about the sampling distribution of the sample mean age is NOT true?
A) The mean of the sampling distribution of sample mean is equal to 24 years.
B) The standard deviation of the sampling distribution of sample mean is equal to 4 years.
C) The shape of the sampling distribution of sample mean is approximately normal.
D) All of these choices are true.

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size. Thus, the standard deviation of the sampling distribution of the sample mean is 4/sqrt(100) = 0.4 years, which is not equal to 4 years. Therefore, choice B is not true. Choices A and C are true because the mean of the sampling distribution of sample mean is equal to the population mean (24 years) and the shape of the sampling distribution of sample mean is approximately normal by the central limit theorem. Therefore, the answer is B.

Question 16

If the population is normal,then $\bar { X }$ is normally distributed for __________________ values of n.

Accepted Answer

The answer of If the population is normal,then \(\bar {...

Question 17

The finite population correction is not needed if the population size is large relative to the sample size.As a rule of thumb,we will treat any population that is at least ____________________ times larger than the sample size as large.

Accepted Answer

The answer of The finite population correction is not needed...

Question 18

For a sample size of 1,the sampling distribution of the mean is normally distributed:
A) regardless of the shape of the population.
B) only if the population values are larger than 30.
C) only if the population is normally distributed.
D) None of these choices.

Accepted Answer

The Central Limit Theorem states that, for sample sizes larger than 30 or when the population distribution is normal, the sampling distribution of the mean will be approximately normally distributed regardless of the shape of the population. However, for sample sizes less than or equal to 30, the sampling distribution of the mean is approximately normally distributed only when the population distribution is normal. Therefore, Choice C is the best option.

Question 19

Heights of Fences
Heights of fences are normally distributed with a mean of 52 inches and a standard deviation of 4 inches.
-{Heights of Fences Narrative} Find the probability that one randomly selected fence is under 54 inches.

Accepted Answer

The answer of Heights of Fences
Heights of fences are normally...

Question 20

As n gets ____________________,the shape of the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.

Accepted Answer

The answer of As n gets ____________________,the shape of the...

Quiz 9: Sampling Distributions

A randomly selected value of $\bar { X }$ is likely to be ____________________ to(than)the mean of the population than is a randomly selected value of X.

Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal.

The width of the sampling distribution of $\bar { X }$ gets ____________________ as the sample size increases.

The variance of the sampling distribution of $\bar { X }$ is ____________________ the variance of the population we're sampling from for all sample sizes.

Which of the following is true about the sampling distribution of the sample mean?

The accuracy of the approximation of $\bar { X }$ with a normal distribution depends on the probability distribution of the and on the sample .

As n gets larger,the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.This phenomenon is due to the ____________________.

For sample sizes greater than 30,the sampling distribution of the mean is approximately normally distributed:

The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.84 pounds.If a sample of 16 fish is taken,what is the standard error of the mean weight?

Because the value of the ____________________ varies randomly from sample to sample,we can regard it as a new random variable created by sampling.

The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.8 pounds.If a sample of 25 roasts yields a mean of 3.6 pounds,what is the Z-score for this sample mean?

The standard error of the mean for a sample of 100 is 25.In order to cut the standard error of the mean in half (to 12.5) we must:

As the number of throws of a fair die increases,the probability that the sample mean is close to (the number)____________________ increases.

Suppose the ages of students in your program follow a positively skewed distribution with mean of 24 years and a standard deviation of 4 years.If we randomly sampled 100 students,which of the following statements about the sampling distribution of the sample mean age is NOT true?

If the population is normal,then $\bar { X }$ is normally distributed for __________________ values of n.

The finite population correction is not needed if the population size is large relative to the sample size.As a rule of thumb,we will treat any population that is at least ____________________ times larger than the sample size as large.

For a sample size of 1,the sampling distribution of the mean is normally distributed:

Heights of Fences Heights of fences are normally distributed with a mean of 52 inches and a standard deviation of 4 inches. -{Heights of Fences Narrative} Find the probability that one randomly selected fence is under 54 inches.

As n gets ____________________,the shape of the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.

Quiz 9: Sampling Distributions

A randomly selected value of Xˉ\bar { X }Xˉ is likely to be ____________________ to(than)the mean of the population than is a randomly selected value of X.

Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal.

The width of the sampling distribution of Xˉ\bar { X }Xˉ gets ____________________ as the sample size increases.

The variance of the sampling distribution of Xˉ\bar { X }Xˉ is ____________________ the variance of the population we're sampling from for all sample sizes.

Which of the following is true about the sampling distribution of the sample mean?

The accuracy of the approximation of Xˉ\bar { X }Xˉ with a normal distribution depends on the probability distribution of the ____________________ and on the sample ____________________.

As n gets larger,the sampling distribution of Xˉ\bar { X }Xˉ becomes increasingly bell shaped.This phenomenon is due to the ____________________.

For sample sizes greater than 30,the sampling distribution of the mean is approximately normally distributed:

The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.84 pounds.If a sample of 16 fish is taken,what is the standard error of the mean weight?

Because the value of the ____________________ varies randomly from sample to sample,we can regard it as a new random variable created by sampling.

The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed,with a mean of 3.2 pounds and standard deviation of 0.8 pounds.If a sample of 25 roasts yields a mean of 3.6 pounds,what is the Z-score for this sample mean?

The standard error of the mean for a sample of 100 is 25.In order to cut the standard error of the mean in half (to 12.5) we must:

As the number of throws of a fair die increases,the probability that the sample mean is close to (the number)____________________ increases.

Suppose the ages of students in your program follow a positively skewed distribution with mean of 24 years and a standard deviation of 4 years.If we randomly sampled 100 students,which of the following statements about the sampling distribution of the sample mean age is NOT true?

If the population is normal,then Xˉ\bar { X }Xˉ is normally distributed for __________________ values of n.

The finite population correction is not needed if the population size is large relative to the sample size.As a rule of thumb,we will treat any population that is at least ____________________ times larger than the sample size as large.

For a sample size of 1,the sampling distribution of the mean is normally distributed:

Heights of Fences Heights of fences are normally distributed with a mean of 52 inches and a standard deviation of 4 inches. -{Heights of Fences Narrative} Find the probability that one randomly selected fence is under 54 inches.

As n gets ____________________,the shape of the sampling distribution of Xˉ\bar { X }Xˉ becomes increasingly bell shaped.

A randomly selected value of $\bar { X }$ is likely to be ____________________ to(than)the mean of the population than is a randomly selected value of X.

The width of the sampling distribution of $\bar { X }$ gets ____________________ as the sample size increases.

The variance of the sampling distribution of $\bar { X }$ is ____________________ the variance of the population we're sampling from for all sample sizes.

The accuracy of the approximation of $\bar { X }$ with a normal distribution depends on the probability distribution of the and on the sample .

As n gets larger,the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.This phenomenon is due to the ____________________.

If the population is normal,then $\bar { X }$ is normally distributed for __________________ values of n.

As n gets ____________________,the shape of the sampling distribution of $\bar { X }$ becomes increasingly bell shaped.