Quality Quiz from Professor Cleary
Last month we looked again at the ongoing struggles of Quinn Quip, who has been trying to understand and implement the binomial central limit theorem. He has learned several lessons related to taking many sample proportions of the size 100 from a population with a mean of .50 and then forming a distribution with these sample proportions. Among the lessons:
- The mean of this distribution of sample proportions will be close to .50;
- The shape would follow a normal distribution;
- The variability of these sample proportions about the population mean of .50 would be
From this, we derived an illustration of what this might look like:
The .40 and .60 are approximately two standard deviations away from the true mean of .50. (For a normal distribution, one would expect to find about 95 percent of the sample proportions to be within this range.) The graphic above shows 20 sample means with extensions representing plus and minus two standard deviations. In only one case does it show an extension that does not include the population mean of .50, implying that the true mean is different from .50. At 95 percent confidence level, one would expect the sample mean with the extension to include the “true mean” 19 times, and not include it one time.
From this, we can introduce the concept of confidence limits. These assure a certain level of confidence that the true proportion will lie within certain limits, when we take a sample proportion and build limits based on this.
The formula to accomplish this is:
Going back to Quinn’s original statement that “Half the people in town—give or take 2 percent—are customers,” at the 95 percent confidence level, is this statement:
Click here for a more complete video explanation
2010 PQ Systems.
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