A normally distributed population has parameters 39.1 and 30.3. If a random sample of size 132 is selected, what is the mean of the distribution of sample means?

Question

29-05-2025
Mathematics

High School

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A normally distributed population has parameters 39.1 and 30.3. If a random sample of size 132 is selected, what is the mean of the distribution of sample means?

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DanielJosephParker DanielJosephParker · Answer 1 · 2024-01-07T02:57:36-05:00

When dealing with sample means from a normally distributed population, one key aspect to remember is that the mean of the distribution of sample means is equal to the mean of the population.

Given:

The population mean ([tex]\mu[/tex]) is 39.1.
A sample size ([tex]n[/tex]) of 132 is selected.

The mean of the distribution of sample means, often denoted as [tex]\mu_{\bar{x}}[/tex], is simply the mean of the population. Therefore, since:

[tex]\mu_{\bar{x}} = \mu = 39.1[/tex]

So, the mean of the distribution of sample means is 39.1.

Why does it work this way?

Central Limit Theorem (CLT): The Central Limit Theorem states that the distribution of the sample means approximates a normal distribution as the sample size becomes larger, regardless of the population's distribution, provided [tex]n \geq 30[/tex]. Here, our sample size, [tex]n = 132[/tex], is sufficiently large.
Consistency: The mean of the sample means ([tex]\mu_{\bar{x}}[/tex]) is always the mean of the population ([tex]\mu[/tex]). This is a fundamental property of sample means derived from the population.

In conclusion, no complex calculations are necessary for this specific question since the mean of the distribution of sample means is directly given by the population mean, 39.1.