Question 1

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year.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 of the mean is:
Standard error = Standard deviation / Square root of sample size
Therefore, in this case, the standard error would be:
$1.2 million / sqrt(100) = $0.12 million

Question 2

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.
A)Distribution is right skewed with mean = 10 minutes and standard error = 0.8 minutes.
B)Distribution is right skewed 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

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution. Here, with a sample size of 100, the distribution of the sample mean is approximately normal. The mean of the sampling distribution is the same as the population mean (10 minutes), and the standard error is calculated as the population standard deviation divided by the square root of the sample size, which is $8 / \sqrt{100} = 0.8$ minutes.

Question 3

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 4

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 5

True or False: 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 answer of True or False: The Central Limit Theorem...

Question 6

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year.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 that the sample mean, X̄ , follows a normal distribution with mean μ =$3.26 million and standard deviation σ/√n= $1.2 million/√100= $0.12 million. We want to find P(X̄ > $3.5 million).

z=(X̄-μ)/σ√n=(3.5-3.26)/(1.2/√100)=1.67

Using a normal distribution table, we get P(Z>1.67)=0.0475, which is the approximate probability of the sample mean exceeding $3.5 million. However, this is the probability of a single sample mean exceeding $3.5 million, not the probability that the population mean exceeds $3.5 million. Therefore, we cannot choose choice D. Since this is a one-tailed test (we are only interested in the probability that the sample mean is greater than $3.5 million), we use the right tail probability, which is 0.0475. Therefore, the best choice is B.

Question 7

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.

Question 8

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

A) The standard error of the mean can be smaller or larger than the standard deviation of the population depending on the sample size and variability of the sample.
B) This is true as the standard error of the mean is inversely proportional to the square root of the sample size.
C) This is also true as the standard error of the mean measures the variability of the sample means from sample to sample, indicating the precision with which the sample mean estimates the population mean.

Question 9

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year.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.
A)approximately 0
B)0.0151
C)0.9849
D)approximately 1

Accepted Answer

We need to use the central limit theorem since the sample size is large enough (n=100) and we do not know the population standard deviation.

z = (x̄ - μ) / (σ / √n)
z = (3.0 - 3.26) / (1.2 / √100)
z = -2.1667

Using a standard normal distribution table or calculator, we find that the probability of obtaining a z-score of -2.1667 or less is approximately 0.0151, which corresponds to answer choice B.

Question 10

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 whose expected value is equal to the parameter it is estimating. This means it is not systematically over- or underestimating the parameter, making it a good estimate of the true value.

Question 11

Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year.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 mean salary exceeding $4.0 million approximately 0.

Question 12

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 disregard the shape of the population when n (sample size) is large. It states that the sampling distribution of the mean of any independent, random variables will be normal or nearly normal, regardless of the shape of the population from which the sample was drawn. This is important because it allows us to make statistical inferences about the population using only a sample, even if the population is not normally distributed.

Question 13

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 statistics, not parameters.

Question 14

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?
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 15

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 enough sample size (n ≥ 30), the distribution of sample means will be approximately normal regardless of the shape of the original population. This is important in statistics because it allows us to make inferences about a population based on a sample, even if we do not know the population's distribution.

Question 16

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 pro-golfers played the course today.Find the probability that the average score of the 36 pro-golfers exceeded 71.

Accepted Answer

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

Question 17

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 equal to σ/√n, not σ.

Question 18

True or False: 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 True or False: The amount of time...

Question 19

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 size of 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 sampling distribution is approximately normal.
D)All of the above are correct.

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 6 ounces.

Question 20

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.
A)right skewed 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)right skewed 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 (standard deviation of the population divided by the square root of the sample size).

Quiz 7: Sampling Distributions

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

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

True or False: 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.

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

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

The standard error of the mean

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

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

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

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

Sampling distributions describe the distribution of

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?

The Central Limit Theorem is important in statistics because

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 pro-golfers played the course today.Find the probability that the average score of the 36 pro-golfers exceeded 71.

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