Question

Suppose the population consists of FIVE individuals and the elements are: S= {3, 6, 9, 12, and 15} Obtain samples of size 3 (use counting rule). Obtain the population mean and variance, sample means and variances of the distribution. Would the mean and variance change if the sample size were to increase? Prepare two excel tables.

a) In an excel table show the various samples (Table-1).

b) Calculate the population mean and variance (Table-1).

c) Calculate the sample mean and standard deviation (Table-1).

d) Why does the population mean and variance differ from sampling distribution of mean and variances?

e) Prepare another table with the following information from the above set S: population mean, population standard deviation and a column with sample means, sample standard deviation and sample errors (Table 2).

f) Explain the statement “as the sample size increases the sample mean tends to and eventually converge upon population mean and population standard deviation”. g) Why would you expect the “sample error to decrease in magnitude as the sample size increases”?

Answer 1

Number of samples of size 3 that can be drawn from a population of 5 = ⁵C₃

Formulas :

Sample mean

Sample varaince

Sample standard deviation

The sample mean usually remain same when sample size increases but the sample varaiance decreases with increase in sample size. The accuracy of a sample mean and varaince enhanced by increasing the sample size.

Now the required table for a), b) and c) is given below.

d) A sample is a part of the population, and sample mean and varaince are alculated from a group of random variables, drawn from the population. Whereas the Population mean and varaince are calculated from the whole population. Usually there is heterogenity in population i.e. there is a varaibility between the population values, and as a result sample mean and varaince differs from the population mean and varaince.