Q1. If the true mean of a population is 16.62, according to the central limit theorem, the mean of the distribution of sample means, for all possible sample sizes n will be:

A)   16.62.

B)   16.62 / √n.

C)   indeterminate for sample with n < 30.

Q2. Suppose the mean debt/equity ratio of the population of all banks in the United States is 20 and the population variance is 25. A banking industry analyst uses a computer program to select a random sample of 50 banks from this population and compute the sample mean. The program repeats this exercise 1000 times and computes the sample mean each time. According to the central limit theorem, the sampling distribution of the 1000 sample means will be approximately normal if the population of bank debt/equity ratios has:

A)    a normal distribution, because the sample is random.

B)    a Student's t-distribution, because the sample size is greater than 30.

C)    any probability distribution.

Q3. Which of the following is NOT a prediction of the central limit theorem?

A)   The variance of the sampling distribution of sample means will approach the population variance divided by the sample size.

B)   The standard error of the sample mean will increase as the sample size increases.

C)   The mean of the sampling distribution of the sample means will be equal to the population mean.

Q4. The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution:

A)   becomes larger.

B)   approaches a normal distribution.

C)   approaches the mean.

Q5. According to the Central Limit Theorem, the distribution of the sample means is approximately normal if:

A)   the underlying population is normal.

B)   the sample size n > 30.

C)   the standard deviation of the population is known.

Q6. The central limit theorem concerns the sampling distribution of the:

A)     sample standard deviation.

B)     population mean.

C)     sample mean.

答案和详解如下：

Q1. If the true mean of a population is 16.62, according to the central limit theorem, the mean of the distribution of sample means, for all possible sample sizes n will be:

A)   16.62.

B)   16.62 / √n.

C)   indeterminate for sample with n < 30.

Correct answer is A)

According to the central limit theorem, the mean of the distribution of sample means will be equal to the population mean. n > 30 is only required for distributions of sample means to approach normal distribution.

Q2. Suppose the mean debt/equity ratio of the population of all banks in the United States is 20 and the population variance is 25. A banking industry analyst uses a computer program to select a random sample of 50 banks from this population and compute the sample mean. The program repeats this exercise 1000 times and computes the sample mean each time. According to the central limit theorem, the sampling distribution of the 1000 sample means will be approximately normal if the population of bank debt/equity ratios has:

A)    a normal distribution, because the sample is random.

B)    a Student's t-distribution, because the sample size is greater than 30.

C)    any probability distribution.

Correct answer is C)

The central limit theorem tells us that for a population with a mean µ and a finite variance σ², the sampling distribution of the sample means of all possible samples of size n will be approximately normally distributed with a mean equal to µ and a variance equal to σ²/n, no matter the distribution of the population, assuming a large sample size.

Q3. Which of the following is NOT a prediction of the central limit theorem?

A)   The variance of the sampling distribution of sample means will approach the population variance divided by the sample size.

B)   The standard error of the sample mean will increase as the sample size increases.

C)   The mean of the sampling distribution of the sample means will be equal to the population mean.

Correct answer is B)

The standard error of the sample mean is equal to the sample standard deviation divided by the square root of the sample size. As the sample size increases, this ratio decreases. The other two choices are predictions of the central limit theorem.

Q4. The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution:

A)   becomes larger.

B)   approaches a normal distribution.

C)   approaches the mean.

Correct answer is B)

As n gets larger, the variance of the distribution of sample means is reduced, and the distribution of sample means approximates a normal distribution.

Q5. According to the Central Limit Theorem, the distribution of the sample means is approximately normal if:

A)   the underlying population is normal.

B)   the sample size n > 30.

C)   the standard deviation of the population is known.

Correct answer is B)

The Central Limit Theorem states that if the sample size is sufficiently large (i.e. greater than 30) the sampling distribution of the sample means will be approximately normal.

Q6. The central limit theorem concerns the sampling distribution of the:

A)     sample standard deviation.

B)     population mean.

C)     sample mean.

Correct answer is C)         

The central limit theorem tells us that for a population with a mean m and a finite variance σ², the sampling distribution of the sample means of all possible samples of size n will approach a normal distribution with a mean equal to m and a variance equal to σ² / n as n gets large.

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