Question 1

The standard error of the mean
A) is never larger than the standard deviation of the population.
B) decreases as the sample size increases.
C) measures the variability of the mean from sample to sample.
D) All of the above.

Accepted Answer

D

Question 2

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players exceeded $3.5 million.
A) Approximately 0
B) 0.0228
C) 0.9772
D) Approximately 1

Accepted Answer

We can use the Central Limit Theorem to approximate the sampling distribution of the sample mean. The mean of the sampling distribution is equal to the population mean, which is $3.26 million. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size, or $1.2 million / sqrt(100) = $0.12 million.

We want to find the probability that the sample mean exceeded $3.5 million, or equivalently, that a standard normal random variable Z = (3.5 - 3.26) / 0.12 = 2,083.33... exceeded a certain threshold. We can use a standard normal distribution table or calculator to find that the corresponding cumulative probability is approximately 0.0228. Thus, the probability that the mean salary of the 100 players exceeded $3.5 million is approximately 0.0228, which corresponds to choice B.

Question 3

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. What was the standard error for the sample mean salary?
A) $0.012 million
B) $0.12 million
C) $12 million
D) $1,200.0 million

Accepted Answer

The formula for standard error is standard deviation divided by the square root of sample size. Thus, SE = $1.2 million / sqrt(100) = $0.12 million.

Question 4

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players exceeded $4.0 million.
A) Approximately 0
B) 0.0228
C) 0.9772
D) Approximately 1

Accepted Answer

The standard error of the mean is $1.2 million / sqrt(100) = $0.12 million. The z-score for $4.0 million is ($4.0 million - $3.26 million) / $0.12 million = 6.17. A z-score of 6.17 is extremely high, making the probability of the sample mean exceeding $4.0 million approximately 0.

Question 5

Suppose the ages of students in Statistics 101 follow a skewed-right distribution with a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 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 equal to 23 years.
B) The standard deviation of the sampling distribution is equal to 3 years.
C) The shape of the sampling distribution is approximately normal.
D) The standard error of the sampling distribution is equal to 0.3 years.

Accepted Answer

The answer of Suppose the ages of students in Statistics...

Question 6

Suppose a sample of n = 50 items is 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 probability distribution with μ = 6 ounces and σ= 2.5 ounces. Which of the following is true about the sampling distribution of the sample mean if a sample of size 15 is selected?
A) The mean of the sampling distribution is 6 ounces.
B) The standard deviation of the sampling distribution is 2.5 ounces.
C) The shape of the sample distribution is approximately normal.
D) All of the above are correct.

Accepted Answer

The answer of Suppose a sample of n = 50...

Question 7

The distribution of the number of loaves of bread sold per day by a large bakery over the past 5 years has a mean of 7,750 and a standard deviation of 145 loaves. Suppose a random sample of n = 40 days has been selected. What is the approximate probability that the average number of loaves sold in the sampled days exceeds 7,895 loaves?

Accepted Answer

The answer of The distribution of the number of loaves...

Question 8

Sales prices of baseball cards from the 1960s are known to possess a skewed-right distribution with a mean sale price of $5.25 and a standard deviation of $2.80. Suppose a random sample of 100 cards from the 1960s is selected. Describe the sampling distribution for the sample mean sale price of the selected cards.
A) skewed-right with a mean of $5.25 and a standard error of $2.80
B) normal with a mean of $5.25 and a standard error of $0.28
C) skewed-right with a mean of $5.25 and a standard error of $0.28
D) normal with a mean of $5.25 and a standard error of $2.80

Accepted Answer

The sampling distribution of the sample mean sale price of the selected cards will be approximately normal due to the Central Limit Theorem, with a mean of $5.25 and a standard error of $0.28, which is the standard deviation of the population divided by the square root of the sample size.

Question 9

Selection of raffle tickets from a large bowl is an example of
A) sampling with replacement.
B) sampling without replacement.
C) subjective probability.
D) None of the above.

Accepted Answer

When raffle tickets are selected from a large bowl, they are not replaced after each selection, meaning that the sample is without replacement. Therefore, the best choice is B. Sampling with replacement (A) would require each ticket to be put back in the bowl after each selection. Subjective probability (C) involves personal opinion and is not applicable in this situation. D is also incorrect as raffle ticket selection is an example of probability.

Question 10

A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) ________ sample.

Accepted Answer

The answer of A sample that does not provide a...

Question 11

The Central Limit Theorem is important in statistics because
A) for a large n, it says the population is approximately normal.
B) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size.
C) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
D) for any sized 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 shape of the population. This is important in statistics because it allows us to use inferential statistics, such as confidence intervals and hypothesis testing, even when the population distribution is unknown or non-normal. Option C correctly states this property of the Central Limit Theorem. Options A, B, and D are partially correct but do not fully capture the key benefit of the theorem. Specifically, option A is true for some populations but not all, and options B and D do not mention the importance of sample size in relation to the theorem.

Question 12

The Central Limit Theorem is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known.

Accepted Answer

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes large, regardless of the shape of the population distribution, provided the population has a finite mean and standard deviation. This makes it a powerful tool in statistics for making inferences about population parameters.

Question 13

For air travelers, one of the biggest complaints is of the waiting time between when the airplane taxis away from the terminal until the flight takes off. This waiting time is known to have a skewed-right distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights.
A) Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutes.
B) Distribution is skewed-right with mean = 10 minutes and standard error = 8 minutes.
C) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8 minutes.
D) Distribution is approximately normal with mean = 10 minutes and standard error = 8 minutes.

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (n=100 is sufficient). The mean of the sampling distribution is the same as the population mean (10 minutes), and the standard error is the population standard deviation divided by the square root of the sample size (8/√100 = 0.8 minutes).

Question 14

Which of the following is true about the sampling distribution of the sample mean?
A) The mean of the sampling distribution is always μ.
B) The standard deviation of the sampling distribution is always σ.
C) The shape of the sampling distribution is always approximately normal.
D) All of the above are true.

Accepted Answer

The mean of the sampling distribution is always equal to the population mean μ. However, the standard deviation of the sampling distribution is equal to σ/√n, where n is the sample size. Additionally, the shape of the sampling distribution is approximately normal when either the population is normal or the sample size is large (by the Central Limit Theorem). Therefore, choice B and C are not always true.

Question 15

The average score of all pro golfers for a particular course has a mean of 70 and a standard deviation of 3.0. Suppose 36 golfers played the course today. Find the probability that the average score of the 36 golfers exceeded 71.

Accepted Answer

The answer of The average score of all pro golfers...

Question 16

If the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic
A) unbiased.
B) minimum variance.
C) biased.
D) random.

Accepted Answer

An unbiased sample statistic is one where the expected value of the statistic is equal to the parameter it is estimating. This means that, on average, the sample statistic will provide an accurate estimate of the population parameter. Therefore, if the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic unbiased.

Question 17

The amount of time it takes to complete an examination has a skewed-left distribution with a mean of 65 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 71 minutes is approximately 0.

Accepted Answer

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

Question 18

Sampling distributions describe the distribution of
A) parameters.
B) statistics.
C) both parameters and statistics.
D) neither parameters nor statistics.

Accepted Answer

Sampling distributions describe the distribution of sample statistics, not parameters.

Question 19

Why is the Central Limit Theorem so important to the study of sampling distributions?
A) It allows us to disregard the size of the sample selected when the population is not normal.
B) It allows us to disregard the shape of the sampling distribution when the size of the population is large.
C) It allows us to disregard the size of the population we are sampling from.
D) It allows us to disregard the shape of the population when n is large.

Accepted Answer

The Central Limit Theorem allows us to assume that the sampling distribution will be approximately normal as the sample size increases, regardless of the shape of the population. This is particularly important when studying large populations because it allows us to make probability statements about the population based on the sample.

Question 20

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

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is not equal to σ, but instead is equal to σ/√n, where σ is the population standard deviation and n is the sample size.

Quiz 7: Sampling and Sampling Distributions

The standard error of the mean

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players exceeded $3.5 million.

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. What was the standard error for the sample mean salary?

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players exceeded $4.0 million.

Suppose the ages of students in Statistics 101 follow a skewed-right distribution with a mean of 23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the following statements about the sampling distribution of the sample mean age is incorrect?

Selection of raffle tickets from a large bowl is an example of

A sample that does not provide a good representation of the population from which it was collected is referred to as a(n) ________ sample.

The Central Limit Theorem is important in statistics because

The Central Limit Theorem is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known.

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

The average score of all pro golfers for a particular course has a mean of 70 and a standard deviation of 3.0. Suppose 36 golfers played the course today. Find the probability that the average score of the 36 golfers exceeded 71.

If the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic

The amount of time it takes to complete an examination has a skewed-left distribution with a mean of 65 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 71 minutes is approximately 0.

Sampling distributions describe the distribution of

Why is the Central Limit Theorem so important to the study of sampling distributions?

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