Answer to Question #160338 in Statistics and Probability for Andrew

Question

Problem Solving

The formula for the standard error (standard deviation of the distribution of sample

means) implies that as the sample size (​n​) increases, the size of the standard error

decreases. Explain the role of the standard error in comparing the sample mean and the

population mean. Use the definitions and concepts on sample means distribution and

standard error and show examples comparing the sample mean and the population mean

of a distribution when the standard error changes in value

Answer 1

Solution:

The formula for the standard error is:

$\sigma_{x'} \approx \frac{s}{\sqrt n}$ , where s - sample standard deviation and n - sample size.

Standard error gives us accuracy of the sample mean, i.e. how sample mean is different from the population mean.It describes how presicely sample mean estimates the true mean of the population.

Here is the example of the sample mean:

Let us consider the mean weight of elefants in 0.x tons. The population consists of 11 animals. Here is the table of weights:

First row is the animal number, the second one - their weights (in 0.x tons)

The first sample is going to be 1-3, second - 1-5, third - 1-9.

The size of the first sample is 3

The size of the second sample is 5

The size of the third population is 9

Here is the table of their respectful sample means

As we can see from the table, as the size of the mean of the population is getting closer to the population mean as we take more samples. Also, the standard error decreases with the growth of the sample sizes.