Why do we calculate s by dividing by n €“ 1, rather than just n?

The reason is that if we divide by n - 1, then s2 is an unbiased estimator of σ2, the population variance. We want to show that s2 is an unbiased estimator of σ2, sigma squared. The mathematical proof that this is true is beyond the scope of an introductory statistics course, but we can use an example to demonstrate that it is. First we will use a very small population that consists only of these three numbers: 1, 2, and 5. You can determine that the population standard deviation, s, for this population is 1.699673 (or about 1.70), as show0n in the TI-84 output. So the population variance, sigma squared, σ2, is 2.888889 (or about 2.89). Now take all possible samples, with replacement, of size 2 from the population, and find the sample variance, s2, for each sample. This process is started for you in the table. Average these sample variances (s2), and you should get approximately 2.88889. If you do, then you have demonstrated that s2 is an unbiased estimator of σ2, sigma squared.

Show your work by filling in the accompanying table and show the average of s2.

Transcribed Image Text:

HORHAL FLORT AUTO REAL RADIRN H ar Stats 2.666666667 Ex2-30 5x=2.081665999 oxm1. 699673171 n=3 minx-1 01-1 T-84 Qutout Sample 2 1,1 0 1, 2 0.707 0.5 1, 5 2.8284 8.0 2, 1 2. 2 Sample 2, 5 5, 1 5, 2 5, 5