1 Discuss in your own words formulas for the mean and the st

1. Discuss in your own words formulas for the mean and the standard deviation for a distribution of sample means?

2. Using the formula for z scores, solve the equation for the mean and also for n

Please in the most simplest way .

And please show how you arrived to the answer Thanks

Solution

Suppose that we draw all possible samples of size n from a given population. Suppose further that we compute a statistic (e.g., a mean, proportion, standard deviation) for each sample. The probability distribution of this statistic is called a sampling distribution.

Variability of a Sampling Distribution

The variability of a sampling distribution is measured by its variance or its standard deviation. The variability of a sampling distribution depends on three factors:

If the population size is much larger than the sample size, then the sampling distribution has roughly the same sampling error, whether we sample with or without replacement. On the other hand, if the sample represents a significant fraction (say, 1/10) of the population size, the sampling error will be noticeably smaller, when we sample without replacement.

Sampling Distribution of the Mean

Suppose we draw all possible samples of size n from a population of size N. Suppose further that we compute a mean score for each sample. In this way, we create a sampling distribution of the mean.

We know the following. The mean of the population (?) is equal to the mean of the sampling distribution (?_x). And the standard error of the sampling distribution (?_x) is determined by the standard deviation of the population (?), the population size, and the sample size. These relationships are shown in the equations below:

?_x = ?      and      ?_x = ? * sqrt( 1/n - 1/N )

Therefore, we can specify the sampling distribution of the mean whenever two conditions are met:

Note: When the population size is very large, the factor 1/N is approximately equal to zero; and the standard deviation formula reduces to: ?_x = ? / sqrt(n). You often see this formula in introductory statistics texts.