Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 357 Step 5: Interpret the result. We are 99% confident that the mean weight �� is between 20.04 and 20.46. Another way to say this is that we are 99% confident that the mean weight �� is in the interval 20.25 ± 0.21. If we were to draw many different samples and use this method to construct the corresponding confidence intervals, then in the long run, 99% of them would cover the true population mean ��. So unless we were very unlucky in the sample we drew, the true mean is between 20.04 and 20.46 ounces. For this 99% confidence interval, the margin of error is 0.2103. For the 90% confidence interval in Example 8.5, the margin of error was only 0.1343. The reason is that for a 90% confidence interval, we used a critical value of 1.645, and for the 99% confidence interval, we must use a larger critical value of 2.576. We can see that if we want to be more confident that our interval contains the true value, we must increase the critical value, which increases the margin of error. There is a trade-off. We would rather have a higher level of confidence than a lower level, but we would also rather have a smaller margin of error than a larger one. So we have to choose a level of confidence that strikes a good balance. The most common choice is 95%. In some cases where high confidence is very important, a larger confidence level such as 99% may be chosen. In general, intervals with confidence levels less than 90% are not considered to be reliable enough to be used in practical situations. Figure 8.8 illustrates the trade-off between confidence level and margin of error. One hundred samples were drawn from a population with mean ��. The center diagram presents one hundred 95% confidence intervals, each based on one of these samples. The confidence intervals are all different, because each sample has a different mean ̄x. The diagram on the left presents 70% confidence intervals based on the same samples. These intervals are narrower because they have a smaller margin of error, but many of them fail to cover the population mean. These intervals are too unreliable to be of any practical value. The figure on the right presents 99.7% confidence intervals. These intervals are very reliable. In the long run, only 3 in 1000 of these intervals will fail to cover the population mean. However, they are wider due to the larger margin of error, so they do not convey as much information. μ μ μ Figure 8.8 Left: One hundred 70% confidence intervals for a population mean, each constructed from a different sample. Although their margin of error is small, they cover the population mean only 70% of the time. This low success rate makes the 70% confidence interval unacceptable for practical purposes. Center: One hundred 95% confidence intervals constructed from these samples. This represents a good compromise between reliability and margin of error for many purposes. Right: One hundred 99.7% confidence intervals constructed from these samples. These intervals cover the population mean 997 times out of 1000. They almost always succeed in covering the population mean, but their margin of error is large.
navidi_monk_elementary_statistics_2e_ch7-9
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