## Confidence Intervals

**Applets**: **Confidence interval for a population mean** ||| **Confidence interval for a population proportion**

** **Use each applet to investigate the meaning of "__% confidence" by clicking on "Update graph" and noticing what percentage of the intervals contain the actual population parameter.

What is the point of this applet?

We often write a conclusion like this: "I have 95% confidence that the population mean falls between 18.3 and 19.5."

What does "95% confidence" mean?

Notice that the sample mean here is (1/2)*(18.3 + 19.5) = 18.7

and the margin of error is (1/2)(19.5 - 18.7) = 0.4

Here are two ways of saying the same thing:

(1) When I take samples of this size in this manner from this population, approximately 95% of the sample means will be within the 0.4 of the population mean.

(2) If I do the process of drawing a random sample from the population, computing the sample statistic, and forming the confidence interval MANY times,

then I can expect that** approximately 95% of these intervals will contain the actual population mean.**

Caution:

This does NOT say that there is a 95% probability that the population mean is between 18.3 and 19.5. As our picture illustrates, the population mean is a fixed number - not random. So our language needs to make clear that it is the endpoints of the interval that are random, not the population mean that is random.

When writing interpretations of "confidence" make sure to refer to repeating the whole process of forming the confidence interval.

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