With the context of sampling distribution, in regression analysis, is the following an appropriate interpretation?
Assumptions :
- X & Y have a linear relationship
- sample size is large enough for Central Limit Theorem to be applicable
Sampling Distribution of the Mean of Y
If all possible samples, m, of sample size n were taken for a specific value of X (i.e. X = h), and mean ($\mu_{y}$) calculated for the Ys observed against those Xs, then :
- The distribution of such means will be normally distributed.
- 95% of the confidence intervals ($ \mu_{y} \: \pm \: 1.96 \sigma$) calculated from such estimated means will contain the true population mean of Y.
- Mean of all the m means will be equal to the overall population mean of Y. (unbiased estimator of a population parameter)
(A statistic is called an unbiased estimator of a population parameter if the mean of the sampling distribution of the statistic is equal to the value of the parameter.)
But since its not practical to take all possible samples of size n, we take 1 sample of size n,
and calculate mean of y.
Confidence Interval for Y
We expect that, $\mu_{y}$ obtained from this one sample is one of those 95% that will yield a confidence interval that will contain the true population mean of Y.
Prediction Interval for Y
The 95% prediction interval (that has a much wider reach than the confidence interval) calculated using the $\mu_{y}$ obtained from this one sample will contain 95% of all the possible population values of Y.
Have I got the interpretation for CI & PI right ?