Answer to Question #174100 in Statistics and Probability for Denisse Bisuña

Question

Consider all samples of size 5 from this population:

2 5 6 8 10 12 13

1. Compute the mean and standard deviation of the population.

2. List all samples of size 5 and compute the mean for each sample.

SAMPLE MEAN

3. Construct the sampling distribution of the sample means.

4. Calculate the mean of the sampling distribution of the sample means. Compare this to the mean of the population.

5. Calculate the standard deviation of the sampling distribution of the sample means. Compare this to the standard deviation of the population.

Answer 1

1. n = 7

$\bar{x} = \frac{2 + 5 + 6 + 8 + 10 + 12 + 13}{7} = 8 \\ sd = \sqrt{ \frac{(2-8)^2 + (5-8)^2 + (6-8)^2 + (8-8)^2 + (10-8)^2 + (12-8)^2 + (13-8)^2}{7-1} } \\ = \sqrt{ \frac{36+9+4+0+4+16+25}{6} } \\ = 3.958$

2.

3.

4. n=21

The mean of the sampling distribution of the sample means $= \frac{\sum Mean_i}{n} = \frac{168}{21} = 8$

The mean of the population is equal to the mean of the sampling distribution of the sample means.

5. The standard deviation of the sampling distribution of the sample means

$= \sqrt{\frac{\sum (Mean_{sd} -Mean_i)^2}{n-1}} \\ = \sqrt{ \frac{18.8}{20} } \\ = 0.9695$

The standard deviation of the sampling distribution of the sample means is less than the standard deviation of the population.