SOLUTION: Hello, Good evening, I need help on my homework. I am having a hard time doing Random Sampling A population consists of (3,8,10,15). Consider all possible sample size 2 that can

Question 1190564: Hello, Good evening, I need help on my homework. I am having a hard time doing Random Sampling
A population consists of (3,8,10,15). Consider all possible sample size 2 that can be drawn with replacement.
-what is the standard deviation of the sampling distribution of means? =
-what is the population mean? =
-what is the population standard deviation? =
-what is the population variance? =
-what is the variance of the sampling distribution of means? =
-what is the mean of the sampling distribution of means? =

A population consists of (3,8,10,15). Consider all possible sample size
2 that can be drawn without replacement.
-what is the variance of the sampling distribution of means? =
-what is the standard deviation of the sampling distribution of means? =
-what is the mean of the sampling distribution of means? =

Answer by chitrank(4) (Show Source): You can put this solution on YOUR website!
Let the first sample be denoted by random variable and the second sample by .
First few notes about and .
---1 They are indepdent RVs. Why? Because sampling happens with replacement. Which means, if I sample , then no matter what really was, the distribution of remains the same.
Now consider the random variable . Do you see why is the sample mean?
Now we can simply apply formulas and theorems to calculate each of the quantities that you have asked.
First, what is mean of the sample-mean? Easy. As you may know, "mean" of a random variable is nothing but an "expectation" operator "E" on the random variable. Therefore mean of the sample means would be which is . Remember is a linear operator? Which gives us . What is , or mean of the population? It is which is . What is ? Well that is also 9 (Remember we are sampling with replacement?)
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