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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

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then we look up from the z table we get

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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

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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?

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the graph above is the sampling distribution of sample means

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2 Answers 2

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.

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we should be able to assume the distribution is normal because of Central Limit Theorem, and in this case $n$ >30 –  user133466 Sep 22 '12 at 20:53
    
but why is the area under the curve to the left of 72.1 less than the area under the curve to the right of 72.1 in the graph above? –  user133466 Sep 22 '12 at 20:55
    
@user133466: We will likely make not too large a mistake if we assume normal distribution. The Central Limit Theorem is a limit theorem when sample size $\to\infty$. That above $30$ we are OK is a rule of thumb, pretty good if the base distribution is not too ugly, but it can be significantly off in some situations. I wanted to be precise. For solving the problem, you are expected to assume normality. –  André Nicolas Sep 22 '12 at 20:59
    
i see, you want to be precise. but can you explain why is the area under the curve to the left of 72.1 is less than the area under the curve to the right of 72.1 in the graph above? –  user133466 Sep 22 '12 at 21:06
    
Suppose your firm is in the business of producing confidence intervals. Consider all situations where the sample mean is say $77.1$. In many of these, the population mean will be greater than $72.1$. It will be $82.1$ just as often as it is $72.1$. –  André Nicolas Sep 22 '12 at 21:18

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!

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