The number of books sold over the course of the three-day book fair were: 365, 303, 401. Assume that samples of size 2 are randomly selected with replacement from this population of three values.
a. List the different possible samples, and find the mean of each of them.
b. Find the mean of your sample means, show your work.
c. Find the mean of your population, show your work.
d. How does the mean of your sample means compare with the mean of your population?
To answer these questions, we'll go step by step.
a. The different possible samples of size 2 that can be selected with replacement from the population of three values (365, 303, and 401) are:
- (365, 365)
- (365, 303)
- (365, 401)
- (303, 365)
- (303, 303)
- (303, 401)
- (401, 365)
- (401, 303)
- (401, 401)
To find the mean of each sample, simply calculate the average of the two values in each sample.
- (365, 365): (365 + 365) / 2 = 365
- (365, 303): (365 + 303) / 2 = 334
- (365, 401): (365 + 401) / 2 = 383
- (303, 365): (303 + 365) / 2 = 334
- (303, 303): (303 + 303) / 2 = 303
- (303, 401): (303 + 401) / 2 = 352
- (401, 365): (401 + 365) / 2 = 383
- (401, 303): (401 + 303) / 2 = 352
- (401, 401): (401 + 401) / 2 = 401
b. To find the mean of your sample means, calculate the average of all the means calculated in step a.
Mean of sample means = (365 + 334 + 383 + 334 + 303 + 352 + 383 + 352 + 401) / 9 = 354
c. To find the mean of the population, calculate the average of the original three values: 365, 303, and 401.
Mean of population = (365 + 303 + 401) / 3 = 356.33 (rounded to two decimal places)
d. The mean of the sample means (354) is slightly lower than the mean of the population (356.33). This is to be expected as the sample means are calculated from smaller subsets of the population and may not perfectly represent the overall population mean. However, with a larger number of samples and larger sample sizes, the mean of the sample means would tend to converge towards the mean of the population.