Example 4.3: Changing the Confidence Level Suppose we change the original problem by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score. Solution To find the confidence interval, you need the sample mean, 
The area to the right of z.025 is 0.025 and the area to the left of z.025 is 1−0.025 = 0.975
Interpretation We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98. Explanation of 95% Confidence Level 95% of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Comparing the results The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider.
Figure 4.1 Comparing the results Summary: Effect of Changing the Confidence Level
Example 4.4: Changing the Sample Size: Suppose we change the original problem to see what happens to the error bound if the sample size is changed. See the following Problem.
Problem Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use n=100 instead of n=36? What happens if we decrease the sample size to n=25 instead of n=36?
Solution B If we decrease the sample size n to 25, we increase the error bound. When n = 25 :
Summary: Effect of Changing the Sample Size
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A confidence interval is a range around a measurement that conveys how precise the measurement is. For most chronic disease and injury programs, the measurement in question is a proportion or a rate (the percent of New Yorkers who exercise regularly or the lung cancer incidence rate). Confidence intervals are often seen on the news when the results of polls are released. This is an example from the Associate Press in October 1996:
Although it is not stated, the margin of error presented here was probably the 95 percent confidence interval. In the simplest terms, this means that there is a 95 percent chance that between 35.5 percent and 42.5 percent of voters would vote for Bob Dole (39 percent plus or minus 3.5 percent). Conversely, there is a 5 percent chance that fewer than 35.5 percent of voters or more than 42.5 percent of voters would vote for Bob Dole. The precise statistical definition of the 95 percent confidence interval is that if the telephone poll were conducted 100 times, 95 times the percent of respondents favoring Bob Dole would be within the calculated confidence intervals and five times the percent favoring Dole would be either higher or lower than the range of the confidence intervals. Instead of 95 percent confidence intervals, you can also have confidence intervals based on different levels of significance, such as 90 percent or 99 percent. Level of significance is a statistical term for how willing you are to be wrong. With a 95 percent confidence interval, you have a 5 percent chance of being wrong. With a 90 percent confidence interval, you have a 10 percent chance of being wrong. A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). A 90 percent confidence interval would be narrower (plus or minus 2.5 percent, for example). What does a confidence interval tell you?he confidence interval tells you more than just the possible range around the estimate. It also tells you about how stable the estimate is. A stable estimate is one that would be close to the same value if the survey were repeated. An unstable estimate is one that would vary from one sample to another. Wider confidence intervals in relation to the estimate itself indicate instability. For example, if 5 percent of voters are undecided, but the margin of error of your survey is plus or minus 3.5 percent, then the estimate is relatively unstable. In one sample of voters, you might have 2 percent say they are undecided, and in the next sample, 8 percent are undecided. This is four times more undecided voters, but both values are still within the margin of error of the initial survey sample. On the other hand, narrow confidence intervals in relation to the point estimate tell you that the estimated value is relatively stable; that repeated polls would give approximately the same results. How are confidence intervals calculated?Confidence intervals are calculated based on the standard error of a measurement. For sample surveys, such as the presidential telephone poll, the standard error is a calculation which shows how well the poll (sample point estimate) can be used to approximate the true value (population parameter), i.e. how many of the people surveyed said they would vote for Dole versus how many people actually would vote for Dole in the election. Generally, the larger the number of measurements made (people surveyed), the smaller the standard error and narrower the resulting confidence intervals. Once the standard error is calculated, the confidence interval is determined by multiplying the standard error by a constant that reflects the level of significance desired, based on the normal distribution. The constant for 95 percent confidence intervals is 1.96. |
What happens to the confidence interval when the confidence level is changed from 95% to 90 %?
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