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A population of values has a normal distribution with [tex]\mu = 38.8[/tex] and [tex]\sigma = 70.9[/tex]. You intend to draw a random sample of size [tex]n = 246[/tex].

What is the mean of the distribution of sample means?

[tex]\mu_{\bar{x}} =[/tex]

Answer :

Final answer:

The mean of the distribution of sample means, when drawing a sample of size 246 from a normally distributed population with a mean of 38.8 and a standard deviation of 70.9, is 38.8.

Explanation:

The question is asking for the mean of the distribution of sample means, which is also known as the sampling distribution of the sample mean, when drawing a random sample from a normally distributed population with a known mean (μ) and standard deviation (σ). According to the Central Limit Theorem, the sampling distribution of the sample mean will be normally distributed with a mean equal to the population mean (μ), regardless of the sample size, as long as the sample size is sufficiently large.

In this case, the population has a mean (μ) of 38.8 and a standard deviation (σ) of 70.9. The sample size (n) is 246, which is generally considered large. In accordance to the Central Limit Theorem, the mean of the distribution of sample means, often represented as μ¯x, would be equal to the population mean. Therefore, the answer is: μ¯x = μ = 38.8

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