Chapter 7, Problem 1P

Briefly define each of the following:

1. a. Distribution of sample means
2. b. Central limit theorem
3. c. Expected value of M
4. d. Standard error of M

To determine

To Define: The distribution of sample means.

Answer to Problem 1P

Distribution of sample means consists of means of all possible samples of fixed size that can be selected from a given population.

Explanation of Solution

Since, it is quite difficult to study the complete population, so sample of fixed size are selected which are representative of population. If all the possible samples of fixed size are listed, they follow certain distributions. Similarly, distribution of sample means consists of sample means of all the possible samples of fixed size that can be selected from a given population.

Conclusion:

Distribution of sample means consists of sample means of all the possible samples of fixed size that can be selected from a given population.

To determine

To Define: Central limit theorem.

Answer to Problem 1P

According to central limit theorem, for any population with mean $\mu$ and standard deviation $\sigma$ , the mean and standard deviation of distribution of sample means of fixed size n are $\mu$ and $\frac{\sigma }{\sqrt{n}}$ respectively as sample size n tend to infinity.

Explanation of Solution

Since, sample means are the representatives of the population means, so as sample size increases, most of the samples piled up closer to the population mean. Therefore, as sample size increases sample means tends to population mean and standard error of sample means decreases. So, this intuition is stated in central limit theorem as:

For any population with mean $\mu$ and standard deviation $\sigma$ , the mean and standard deviation of distribution of sample means of fixed size n are $\mu$ and $\frac{\sigma }{\sqrt{n}}$ respectively as sample size n tend to infinity.

Conclusion:

According to central limit theorem, for any population with mean $\mu$ and standard deviation $\sigma$ , the mean and standard deviation of distribution of sample means of fixed size n are $\mu$ and $\frac{\sigma }{\sqrt{n}}$ respectively as sample size n tend to infinity.

To determine

To Define: Expected value of M.

Answer to Problem 1P

The expected value of M is the average of means for all the possible samples of the fixed size which can be selected from given population.

Explanation of Solution

The sample means are the representative of the population mean from which samples have been drawn. The average of means for all possible samples of fixed size takes all population units into the consideration and equals to population mean. Therefore, the expected value of M is the average of means for all the possible samples of the fixed size which can be selected from given population and is equals to the population mean.

Conclusion:

The expected value of M is the average of means for all the possible samples of the fixed size which can be selected from given population.

To determine

To Define: Standard error of M.

Answer to Problem 1P

Standard error of M measures the average distance of M from the population mean and is equals to $\frac{\sigma }{\sqrt{n}}$.

Explanation of Solution

Since, distribution of sample means consists of means for all the possible samples of fixed size n from the given population. So, standard error of M measures the average distance of M from the population mean. In other words, standard error of M is the standard deviation of the distribution of the sample means and is equals to $\frac{\sigma }{\sqrt{n}}$.

Conclusion:

Standard error of M is the standard deviation of the distribution of the sample means and is equals to $\frac{\sigma }{\sqrt{n}}$.