Question

A population has a mean of 50 and a standard deviation of \(6 .\) (a) What are the mean and standard deviation of the sampling distribution of the mean for \(\mathrm{N}=\) \(16 ?\) (b) What are the mean and standard deviation of the sampling distribution of the mean for \(\mathrm{N}=20 ?\)

Short answer

The mean is 50 for both, with standard deviations of 1.5 for N=16 and 1.34 for N=20.

Step-by-step solution

Understanding the Sampling Distribution Mean

The mean of the sampling distribution of the sample mean is equal to the mean of the population. Therefore, no matter what the sample size is, the mean of the sampling distribution is 50.

Calculating the Standard Error for N=16

The standard deviation of the sampling distribution (standard error) is calculated by dividing the population standard deviation by the square root of the sample size. For \(N=16\) this is \(\frac{6}{\sqrt{16}} = \frac{6}{4} = 1.5\).

Calculating the Standard Error for N=20

Similarly, for \(N=20\), the standard deviation of the sampling distribution is \(\frac{6}{\sqrt{20}} = \frac{6}{4.47} \approx 1.34\).

Key concepts

Understanding the Mean

The mean is a central concept in statistics, representing the average of a data set. When discussing the sampling distribution of the mean, it’s essential to know that the mean of the sampling distribution is the same as the mean of the entire population. This holds true regardless of the sample size. In our original exercise, the population mean is given as 50. Thus, the mean of the sampling distribution, whether the sample size is 16 or 20, remains 50. This encourages consistency across observations that are central to grasping the concept of mean in probability and statistics.

Exploring Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. When dealing with a population, the standard deviation gives us insight into how much individual data points deviate from the population mean. In the context of our exercise, the population standard deviation is 6. This number reflects the extent to which individual observations typically differ from the mean of 50. With a smaller standard deviation, points tend to be closer to the mean, indicating less variance in the data. Understanding standard deviation is crucial for interpreting how data is spread out or clustered within a distribution.

Deciphering Standard Error

The standard error is a measure of how much the mean of the sample is expected to differ from the population mean. It provides insights into the precision of the sample mean as an estimate of the population mean. Computed as the standard deviation of the population divided by the square root of the sample size, the standard error accounts for the impact of sample size on variability.

• For a sample size of 16, the standard error is calculated as \( \frac{6}{\sqrt{16}} = 1.5 \).
• If the sample size increases to 20, the standard error changes to \( \frac{6}{\sqrt{20}} \approx 1.34 \).

This decrease in the standard error with a larger sample size highlights a crucial point: larger samples tend to provide a more accurate estimate of the population mean, evidenced by a smaller spread of the sampling distribution mean around the population mean.