Estimating a Population Variance, Chi-Square Distribution Properties
Estimating a Population Variance
Chi-square Distribution Table
Constructing a confidence interval for σ2
Estimating a Population Variances
Finding a confidence interval estimate of a population standard deviation or variance. We introduce the chi-square distribution to do this. In a normally distributed population with variance σ2, assume that we randomly select independent samples of size n and, for each sample, compute the sample variance S2.
where n is the sample size; S2 is the sample variance; and σ2 is the population variance has a sampling distribution called the chi-square distribution.
It is not symmetric, unlike the normal and Student t-distributions
Values of chi-square can be positive or zero, but not negative that It is different for each number of degrees of freedom.
Because chi-square is not symmetric, we cannot apply a confidence interval formula of S2±E. We must do separate calculations for the upper and lower confidence interval limits. For the population variance:
For the population standard deviation:
σ is the population of standard deviation
s sample standard deviation
n is the number of sample values
σ2 is population variance
S2 is the sample variance
XL2 is left tail critical value of x2
XR2is right tail critical value of x2
Critical values of the chi-square distribution are contained in chi-square tables, Degrees of freedom, as with the Student t-Distribution, degrees of freedom is n-1.
A simple random sample of ten voltage levels of a typical family house is obtained.
Construction of a confidence interval for the population standard deviation σ requires the left and right critical values of χ2 corresponding to a confidence level of 95% and a sample size n = 10.Find the critical value of χ2 separating an area of 0.025 in the left tail, and find the critical value of χ2 separating an area of 0.025 in the right tail.
df=n-1 and critical values 0.025 where confidence level estimation :1- 0.0025=0.975
on the table you will find 2,7
University of Exeter LibGuide is licensed under CC BY 4.0
