ch7_slides - INFERENCES ABOUT POPULATION VARIANCES...

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Unformatted text preview: INFERENCES ABOUT POPULATION VARIANCES Estimation and Tests for a Single Population Variance • Recall that the sample variance s 2 = ∑ ( y- ¯ y ) 2 / ( n- 1) is the point estimate of σ 2 . • For tests and confidence intervals about σ 2 we use the fact that the sampling random variable ( n- 1) S 2 /σ 2 = χ 2 has the Chi-square Distribution with n- 1 degrees of freedom or df , when the sample is from a normal population. 1 • The chi-square distribution is nonsymmetric. • Like the Student’s t distribution, there is a different curve for each sample size n i.e., for each value of df . • Percentiles of the chi-square distribution are given in Table 7. • Plots of the chi-square distribution for df = 5 , 15 , and 30 are shown in Fig. 7.3 • The distribution appear to be more skewed for smaller values of df and become more symmetric as df increases. • Because the chi-square distribution is nonsymmetric, the percentiles for probabilities at both ends needs to be tabulated. 2 3 A 100(1- α ) % Confidence Interval for σ 2 This has the form: ( n- 1) s 2 χ 2 U < σ 2 < ( n- 1) s 2 χ 2 L where • Since df = n- 1 look up χ 2 n- 1 percentiles. • χ 2 L is the lower-tail value with area α/ 2 to the left. • χ 2 U is the upper-tail value with area α/ 2 to the right. • The confidence interval for the standard deviation σ is found by taking square roots of both end points of above. 4 Example 7.1 The normal probability plot of the data (see text book) appear to show that the sample is from a normal distribution. From the data n = 30 , ¯ y = 500 . 453 , s = 3 . 433 were calculated....
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This note was uploaded on 01/23/2010 for the course STAT 213 taught by Professor Hao during the Spring '10 term at Internet2.

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ch7_slides - INFERENCES ABOUT POPULATION VARIANCES...

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