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Suppose the brain volume for adult women is about 1,100 cc, with a standard deviation of 75 cc. Then consider a sample of 144 random women taken from this population, with an unspecified sample mean. Determine the approximate 10th percentile of the distribution of the distribution of sample means of the 144 women?


The solution to this problem hinges on the determination that the standard deviation of the sample mean is $75/12 = 6.25$. But why is this true? How could we possibly justify this? Where does the number, 12, come from?

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The number 12 appears because there was a sample of 144 women, and $\sqrt{144}=12$.

This is explained by the Central Limit Theorem. In general, if $X$ is a random variable then, if you take $n$ samples of $X$ a bunch of times, the mean of those samples, $\bar{X}$ will be normally distributed with $\bar{X} \sim \mathcal{N}\left(\mu_{_X},\dfrac{\sigma_{_X}}{\sqrt{n}}\right)$ where $\mu_{_X}$ and $\sigma_{_X}$ are the population mean and standard deviation, respectively.

To see why this makes sense, I would recommend watching this fantastic video by 3Blue1Brown on the topic: But what is the Central Limit Theorem?

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  • $\begingroup$ Ok, I think this makes sense. But I want to clarify a couple things. Am I correct to interpret the question as stating that we are dealing with multiple samples of the same size (144)? And do the samples all need to be the same size for the Central Limit Theorem to apply? $\endgroup$
    – Stanley Yu
    Apr 10, 2023 at 21:45
  • $\begingroup$ Oh, and can we assume that this formula for the mean of the sample holds for other statistics? For example, the mean and standard deviations of the variances of the samples? $\endgroup$
    – Stanley Yu
    Apr 10, 2023 at 21:55
  • $\begingroup$ @StanleyYu You don't need to take multiple samples for the CLT to apply, and you can take samples of any, sufficiently large, size. The theorem says that if you take a sample of size $n$ from a population with known mean and st. dev., then whatever the average of that sample turns out to be, it will fall under the normal distribution that I gave in my answer. When I said "take $n$ sample a bunch of times" I meant that, in order to see the normal distribution to take shape, you would need to take many samples of that same size. $\endgroup$
    – MikeV
    Apr 12, 2023 at 16:45
  • $\begingroup$ @StanleyYu To answer your second question. From my understanding, the CLT applies only to the sample mean and can not be used to glean any other statistics. $\endgroup$
    – MikeV
    Apr 12, 2023 at 16:45

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