Confusion on the meaning of confidence interval  The standard deviation of test scores on a certain achievement test is  10.9. A random sample of 60 scores on this test had a mean of  72.1. Based on this sample, find a 95% confidence interval for the true mean of all scores.
The formula for confidence interval in this case is

then we look up from the z table we get

Confusion starts here: 95% confidence interval means there's 47.5% to the left 47.5% to the right if the sample mean is in the middle

Suppose that our sample mean, 72.1, is below the actual mean of the population. We have the following graph, just by looking at the graph, you can tell there's more area under the curve to the right of 72.1 than there is area to the left How is it possible that the area to the left is 47.5% and area to the right is 47.5% ?? im confused.....How can we say that confidence interval is equal distant $\approx$2.758 (distance to the left and distance to the right) from the sample mean?

the graph above is the sampling distribution of sample means
 A: Why not? We are assuming that scores are (exactly) normally distributed. The error made in this case (distribution of scores is not really normal) is probably not too bad. (Maybe not too good, either, the sample size is not large.)
We have no more reason to think that the true mean is above the sample mean than to think the true mean is below the sample mean. So the cases that you discuss, where the sample mean is below the true mean, are cancelled by the cases where the sample mean is above the true mean.  the mean value of the sample mean is $\mu$, the true mean, and the distribution of the sample mean is symmetrical about the true mean. 
A: The confidence interval is centered at the sample mean not the population mean.  Generally the sample mean will not equal to the population mean.  So the interval will always be centered to the left or right of the population mean and will not be at the cneter of the sampling distribution for the mean.  What the confidence interval does is contain the population in 95% of the cases in which it is computed and in 5% of those cases the population mean will actually lie outside the interval.  So not only is it not at the center of the interval but sometimes it is actually all the way outside of the interval! 
