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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
Let the first sample be denoted by random variable X_1 and the second sample by X_2.
First few notes about X_1and X_2.
---1 They are indepdent RVs. Why? Because sampling happens with replacement. Which means, if I sample X_1, then no matter whatX_1 really was, the distribution of X_2 remains the same.
Now consider the random variable X_3=%28X_1%2BX_2%29%2F2. Do you see why X_3 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 E%28X_3%29 which is E%28%28X_1%2BX_2%29%2F2%29. Remember E is a linear operator? Which gives us E%28X_3%29=%28E%28X_1%29%2BE%28X_2%29%29%2F2. What is E%28X_1%29, or mean of the population? It is %283%2B8%2B10%2B15%29%2F4 which is 36%2F4=9. What is E%28X_2%29? Well that is also 9 (Remember we are sampling with replacement?)