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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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