Lesson Central Limit Theorem

Randomization for Generalization
Remember, sampling is an important tool for determining the characteristics of a population. We usually don't know the population's parameters (mean, standard deviation, etc.), but often want reliable estimates of them. Ensuring random (representative) sampling free of bias and sampling errors is important. Some sources of error can be accounted for in the experimental design (blind, double blind, latin square, etc.). An important rule to remember is:
No randomization, no generalization.
What this means is, your results can not be generalized if proper randomization techniques did not occur in your sampling. Many masters degree students have visited their statistician AFTER collecting their data and discovered many months or years were wasted due to poor experimental design.
The Central Limits Theorem
A very important and useful concept in statistics is the Central Limit Theorem. There are essentially three things we want to learn about any distribution: 1) The location of its center; 2) its width, 3) and how it is distributed. The central limit theorem helps us approximate all three.
Central Limit Theorem: As sample size increases, the sampling distribution of sample means approaches that of a normal distribution with a mean the same as the population and a standard deviation equal to the standard deviation of the population divided by the square root of n (the sample size).
Stated another way, if you draw simple random samples (SRS) of size n from any population whatsoever with mean and finite standard deviation , when n is large, the sampling distribution of the sample means is close to a normal distribution with mean and standard deviation / (n). This normal distribution is often denoted by: N(, / (n)).

Questions:
Ourtown Health Department reported that the height of women in the city is approximately normally distributed with a mean of 5 feet, 4 inches (i.e., 64 inches) and a standard deviation of 3 inches.
Suppose we select a random sample of five women from our school, measure the height of each, and calculate the sample mean. If we wished to know whether the height of women at our school is typical of the height of women in Ourtown, how should we compare our sample data to information we have about the Ourtown population distribution?
We should compare the sample mean to the population distribution of Ourtown women as we do with an individual score.
We should compare each individual score in our sample, one at a time, to the population of individual scores.
There really isn't a way to make any worthwhile comparison.
We should compare this sample mean to a sampling distribution of all possible means for samples of 5 women from the population of women in Ourtown
Answer:
We should compare this sample mean to a sampling distribution of all possible means for samples of 5 women from the population of women in Ourtown
We have to compare our sample mean with a distribution that is made up of all possible sample means (a sampling distribution of means). We want to compare our mean to a distribution of all possible sample means drawn from the population of interest. To compare our sample mean with a distribution of individual scores would be comparing apples to pickles.