Question 1

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 2

The variance of the sampling distribution of  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 3

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

To find the Z-score, we use the formula:

$Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Where $\bar{X}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Substituting the given values, we get:

$Z = \frac{3.6 - 3.2}{\frac{0.8}{\sqrt{25}}} = \frac{0.4}{0.16} = 2.5$

Therefore, the Z-score for this sample mean is 2.5. The best choice is B.

Question 4

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 5

If the population is normal, then  is normally distributed for __________________ values of n.

Accepted Answer

The answer of If the population is normal, then ...

Question 6

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 7

As n gets larger, the sampling distribution of  becomes increasingly bell shaped.This phenomenon is due to the ____________________.

Accepted Answer

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

Question 8

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 9

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 a sample size of 1, the sampling distribution of the mean is normally distributed only if the population is normally distributed. Therefore, choice C is the best answer.

Question 10

As n gets ____________________, the shape of the sampling distribution of  becomes increasingly bell shaped.

Accepted Answer

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

Question 11

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 12

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

Accepted Answer

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

Question 13

The accuracy of the approximation of  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 ...

Question 14

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 15

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 16

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 17

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 sample mean is not equal to the population standard deviation, but is equal to the population standard deviation divided by the square root of the sample size. Thus, the standard deviation of the sampling distribution of sample mean would be 0.4 (i.e. 4/√100), which is smaller than the standard deviation of the population. Therefore, statement B is not true. Statements A and C are true as the mean of the sampling distribution of sample mean is the same as the population mean, and the shape of the sampling distribution of sample mean is approximately normal by the central limit theorem.

Question 18

The width of the sampling distribution of  gets ____________________ as the sample size increases.

Accepted Answer

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

Question 19

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  ?
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

The answer of Suppose that 100 items are drawn from...

Question 20

A randomly selected value of  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  is...

Quiz 9: Sampling Distributions

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:

The variance of the sampling distribution of is ____________________ the variance of the population we're sampling from for all sample sizes.

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?

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.

If the population is normal, then is normally distributed for __________________ values of n.

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.

As n gets larger, the sampling distribution of becomes increasingly bell shaped.This phenomenon is due to the ____________________.

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?

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

As n gets ____________________, the shape of the sampling distribution of becomes increasingly bell shaped.

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

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

The accuracy of the approximation of with a normal distribution depends on the probability distribution of the and on the sample .

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

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

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.

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?

The width of the sampling distribution of gets ____________________ as the sample size increases.

A randomly selected value of is likely to be ____________________ to(than) the mean of the population than is a randomly selected value of X.