Discuss the three primary points of the central limit theorem. Please do not cop

Question

Discuss the three primary points of the central limit theorem. Please do not copy the information directly from the textbook. Rather, provide an explanation in your own words. One point has to do with the shape of sampling distribution. Another point has to do with the relationship between the mean of a sampling distribution to mu (population mean). The last point has to do with the relationship between the population standard deviation and the sampling distribution standard deviation. Again, use the example of ages of all people in the United States to make the point. You may make up a population standard deviation for these ages.

Explanation / Answer

The Central Limit Theorem consists of three statements:

[1] The mean of the sampling distribution of means is equal to the mean of the population from which the samples were drawn.

[2] The variance of the sampling distribution of means is equal to the variance of the population from which the samples were drawn divided by the size of the samples.

or we can say, the standard deviation of the sampling distribution of means is equal to the standard deviation of the population from which the samples were drawn divided by the square root of the size of the samples.

[3] If the original population is distributed normally (i.e. it is bell shaped), the sampling distribution of means will also be normal. If the original population is not normally distributed, the sampling distribution of means will increasingly approximate a normal distribution as sample size increases. (i.e. when increasingly large samples are drawn)

For example:-

The United States has a total resident population of 325,120,392 (3.25 million), making it the third most populous country in the world.

The average life expectancy is 79.59 years or 80 years (rounding to the wohle number.)

Now, we consider the population mean it is 80 years, as given above, and now if we consider sample of say 100 people, the mean will be 80 years as well, as per the firs point above,

If we talk about the standard deviation, let say the standard deviation of life expectancy for population, it is 1.86 years or 2 years (rounding to the whole number).

Then the standard deviation for the sample will be, 2/sqrt(100) = 0.02 years

And if we consider the original population is distributed normally (i.e. it is bell shaped), the sampling distribution of means will also be normal. If the original population is not normally distributed, the sampling distribution of means will increasingly approximate a normal distribution as sample size increases. (i.e. when increasingly large samples are drawn)

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