Population Statistics: The Variability Of Sample Populations

Submitted By Rick-Yeh

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VAR of the sample average depends on 2 things: a) VAR of the sample population ( and b) sample size ((n).

The variability of sample average is decreasing with larger sample size (larger value of n).
As gets smaller the sample size (n) gets larger

All else being equal, the larger the sample size (n), the steeper/taller/narrower the normal curve (tells us how close we are to the true value).

Variability of sample average is larger when the population variance is larger. Larger population variance means our individual draws of the X’s are more spread out.
Unbiased – since the expected value of is equal to the thing we are trying to estimate, we say that is an unbiased estimate of the population mean ().
The sample Xi must be independent, which means covariance = 0 in .
= E(Xi) = expected mean
Let X1, X2, … Xn ~ N(, ) iid, then ~ N(,( /n)) This is sample average for normal distribution.

------- This is for sample average distribution calculation
------------ 95% confidence interval
------------ 68% confidence interval

We denote standard error as , and using the formula = / where n = sample size (find sample mean’s standard deviation to ultimate find 95% confidence interval).
Confidence Intervals answer the basic questions about what our parameter is and how sure we are of the parameter.
Small interval, we can know a lot about it.
Larger interval – we don’t know much.
Means if we take 100 samples and create 100 difference

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