{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ch7_slides

# ch7_slides - INFERENCES ABOUT POPULATION VARIANCES...

This preview shows pages 1–6. Sign up to view the full content.

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 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. A 99% confidence interval for σ 2 is computed as follows: Since α = . 1 , α/ 2 = . 005 and 1 - α/ 2 = . 995 , we compute χ 2 L = χ 2 0 . 995 , 29 = 13 . 12 χ 2 U = χ 2 0 . 005 , 29 = 52 . 34 Thus the endpoints for the required C.I. are, respectively,: ( n - 1) s 2 χ 2 U = 29(3 . 433 2 ) 52 . 34

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}