Question 1

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

Accepted Answer

The mean of the sampling distribution of the sample mean is always equal to the population mean (μ). Option B is not true as the standard deviation of the sampling distribution is equal to σ/√n (where σ is the population standard deviation and n is the sample size), and it is not always equal to σ. Option C is also not always true as the shape of the sampling distribution can be approximately normal under certain conditions (such as when the sample size is large enough), but it is not always the case.

Question 2

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

biased

Question 3

Suppose the ages of students in Statistics 101 follow a right skewed 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?

Accepted Answer

B

Question 4

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

Accepted Answer

If the expected value of a sample statistic is equal to the parameter it is estimating, then we call it an unbiased estimator.

Question 5

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

Accepted Answer

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

Question 6

The Central Limit Theorem is important in statistics because

Accepted Answer

The Central Limit Theorem states that for a large enough sample (n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. Therefore, choice C is the best answer.

Question 7

Sales prices of baseball cards from the 1960s are known to possess a right skewed 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.

Accepted Answer

The sampling distribution of the sample mean is approximately normal, regardless of the shape of the population distribution, due to the Central Limit Theorem. The mean of the sampling distribution is equal to the mean of the population, which is $5.25. The standard error of the mean is equal to the standard deviation of the population divided by the square root of the sample size, which is $2.80 / sqrt(100) = $0.28.

Question 8

Sampling distributions describe the distribution of

Accepted Answer

Sampling distributions describe the distribution of statistics, which are numerical summaries of the sample data. Parameters are numerical values that describe the population, and are not affected by sampling variability.

Question 9

Suppose a sample of n = 50 items is selected 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?

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 6 ounces. The standard deviation of the sampling distribution is not 2.5 ounces; it is the population standard deviation divided by the square root of the sample size (σ/√n). The shape of the sampling distribution is approximately normal only if the sample size is large enough (typically n ≥ 30) or if the population distribution is normal. Since n = 15, the shape may not be approximately normal unless the population distribution is normal.

Question 10

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 right
Skewed 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.

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 large). 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 11

The distribution of the number of loaves of bread sold per week 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 weeks has been selected. What is the approximate probability that the mean number of
loaves sold in the sampled weeks exceeds 7,895 loaves?

Accepted Answer

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

Question 12

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. 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.

Accepted Answer

The standard error of the mean is $1.2 million / sqrt(100) = $0.12 million. The z-score for $3.5 million is ($3.5 million - $3.26 million) / $0.12 million = 2. The probability of a z-score greater than 2 is 0.0228.

Question 13

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

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 14

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. Suppose a sample of 100 major league players was taken. Find the
Approximate probability that the mean salary of the 100 players was no more than $3.0 million.

Accepted Answer

The standard error of the mean is $1.2 million / sqrt(100) = $0.12 million. The z-score for a mean salary of $3.0 million is ($3.0 million - $3.26 million) / $0.12 million = -2.17. Using the standard normal distribution table, the probability of a z-score less than -2.17 is approximately 0.0151.

Question 15

The standard error of the mean

Accepted Answer

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

Question 16

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 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

The standard error (SE) is calculated by dividing the standard deviation by the square root of the sample size: SE = $1.2 million / √100 = $0.12 million.

Question 19

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. 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.

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 - $3.26) / $0.12 = 6.17. A z-score of 6.17 is extremely high, making the probability of exceeding $4.0 million approximately 0.

Question 20

The amount of time it takes to complete an examination has a left skewed
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...

Quiz 7: Sampling Distributions

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

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

Suppose the ages of students in Statistics 101 follow a right skewed 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?

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

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

The Central Limit Theorem is important in statistics because

Sampling distributions describe the distribution of

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. 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.

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

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. Suppose a sample of 100 major league players was taken. Find the Approximate probability that the mean salary of the 100 players was no more than $3.0 million.

The standard error of the mean

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

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past. 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 a certain year in the past. 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.

The amount of time it takes to complete an examination has a left skewed 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.