If the mean of the sampling distribution of the means is 7.25 what is the mean of the population

Sample statistics are random variables because they vary from sample to sample. As a result, sample statistics have a distribution called the sampling distribution. The video below demonstrates the construction of a sampling distribution for a known population proportion using StatKey (http://www.lock5stat.com/StatKey/index.html). StatKey is a free online application that we will be using throughout the course.

An important aspect of a sampling distribution is the standard error (SE). The standard error is the standard deviation of a sampling distribution.  For a single categorical variable this may be referred to as the standard error of the proportion. For a single quantitative variable this may be referred to as the standard error of the mean. If a sampling distribution is constructed using data from a population, the mean of the sampling distribution will be approximately equal to the population parameter.

Sampling Distribution Distribution of sample statistics with a mean approximately equal to the mean in the original distribution and a standard deviation known as the standard error

Standard Error Standard deviation of a sampling distribution

Using StatKey to Construct a Sampling Distribution Given a Known Population Proportion

Note that this method of constructing a sampling distribution requires that we have population data. In most cases we do not know all of the population values. If we did, then we wouldn't need to construct a confidence interval to estimate the population parameter! In those cases we can use bootstrapping methods which you will see in the next section. 

As you look through the following examples, note that when the sample size is large the sampling distribution is approximately symmetrical and centered at the population parameter. 

7.3 Mean and Standard Deviation of307xIf we calculate the mean and standard deviation of the 10 values oflisted in Table 7.3,we obtain the mean,and the standard deviation,ofAlternatively, we can calculate themean and standard deviation of the sampling distribution oflisted in Table 7.5. These willalso be the values ofandFrom these calculations, we will obtainand(see Exercise 7.25 at the end of this section).The mean of the sampling distribution ofis always equal to the mean of the population.xsx3.30mx80.60sx.mxxx.sx,mx,xMean of the Sampling Distribution ofThemean of the sampling distribution ofis alwaysequal to the mean of the population. Thus,mxmxxHence, if we select all possible samples (of the same size) from a population and cal-culate their means, the mean () of all these sample means will be the same as the mean() of the population. If we calculate the mean for the population probability distribution ofTable 7.2 and the mean for the sampling distribution of Table 7.5 by using the formula learnedin Section 5.3 of Chapter 5, we get the same value of 80.60 forand(see Exercise 7.25).The sample mean,is called anestimatorof the population mean,. When the expectedvalue (or mean) of a sample statistic is equal to the value of the corresponding population parame-ter, that sample statistic is said to be anunbiased estimator. For the sample meanHence,is an unbiased estimator of. This is a very important property that an estimator should possess.However, the standard deviation,ofis not equal to the standard deviation,, of thepopulation distribution (unlessn1). The standard deviation ofis equal to the standard de-viation of the population divided by the square root of the sample size; that is,This formula for the standard deviation ofholds true only when the sampling is done eitherwith replacement from a finite population or with or without replacement from an infinite pop-ulation. These two conditions can be replaced by the condition that the above formula holdstrue if the sample size is small in comparison to the population size. The sample size is consid-ered to be small compared to the population size if the sample size is equal to or less than 5%of the population size—that is, ifIf this condition is not satisfied, we use the following formula to calculatewhere the factoris called the finite population correction factor.

How do you find the mean of the sampling distribution of sample means?

When the mean of a number is 18 What is the mean of the sampling distribution?

What happens to the mean of the sampling distribution of the sample means when the sample size increases?