3/06/95 conf CONFIDENCE INTERVALS AND HYPOTHESIS TESTING FOR VARIANCES1) A Confidence Interval for the Population VarianceThe Chi-squared 2 distribution refers to the distribution of a sum of z2s, that is sums like2122232 2 z z z znwhere z is a N ( , )0 1 variable, that is, it is normal with a mean of zero and a standard deviation of one. The Chi-squared distribution is tabulated according to degrees of freedom (DF) and has mean = DF, variance = 2DF.For example, if a chi-squared statistic has seven degrees of freedom, its mean is 7 and its variance is 14.To find a confidence interval for a population variance, we must first estimate the sample variance, 112222nxnxnxxs , where n is the size of the sample. If x is normally distributed, zx x will be have the standardized normal distribution with a mean of zero and a standard deviation of one 1,0~ Nz. The sum of these z's squared will be a chi-squared, 2222221snxxz, and will have a 2 distribution with n -1 degrees of freedom. If we are looking for a 95% confidence interval for a variance we can observe that since the above ratio has a 2 distribution, there must be two numbers, . .97520252 and , that together cut off an interval about the mean that contains 95% of the probability. We can indicate this by saying %9512025.222975.snP. But if the expression in brackets is true 95% of the time, so is its inverse, 2975.222025.111sn . And if this is true, it is also true after being multiplied through by 21 sn . So we get 2975.222025.211snsn as an interval for the variance, or , more generally, for confidence level 1, 2212222211 snsn . For example, if n 31 ands 1000, our degrees of freedom are n 1 30, so that if the confidence level is 95%, we look up .( ).( ). .0252 309752 3046 979 16 791 and . Substituting these into the formula we get 791.16100030999.46100030222 , which can be reduced to 22210007867.110006386.0 . If we want a confidence interval for the standard deviation instead, we can take the square root of both sides of the equation and say 1000337.11000799.0 or 799 1337 .2)Finding values for 2On most tables of the chi-squared distribution the right hand critical values of 2 appear alone. For example for .025 and 5 degrees of freedom the appropriate critical value of chi-squared is 12.8. This means that any value of 2 above 12.8 is in the right-hand tail of the distribution where only 2.5% of the probability lies. A value of 2 above 12.8 would cause us to reject a null hypothesis for a 2.5% one-sided test or a 5% two-sided test. Unfortunately many chi-squared tables give only these upper critical values and leave out lower critical values like .9752 , so a table with both upper and lower critical values is included with this document.However, this table does not show all values of 2 above 30 degrees of freedom. For more than 30 degrees of freedom the normal distribution is used to approximate the chi-squared distribution, most commonly by setting z DF 2 2 12 , where DF n 1 . For example, if 110100011001001 then 101 and 1000,110022222snns, then, if this value of 2 is substituted into the above formula for z, we get 7257.01067.148324.14199220110021102 z .An example of the use of this appears under "Hypothesis Testing for Variances - One Sample."This formula is also substituted into the confidence interval formula to give the following approximate formula for a confidence interval for a standard deviation:s DFz DFs DFz DF22222 2 , which could also be written as s DFDF z222 . For example, if 28.25=798=1-4002=2 ,05. and 1000 ,400 DFsn , sothat 1075.935or 96.125.2825.281000 If a confidence interval for the variance is required, the numbers on both sides of the interval can be squared. This interval should only be used when the values of 2 for the appropriate degrees of freedom are above those available on the chi-squared table.3) Hypothesis Testing for Variances - One SampleSuppose that we want to test the statement that the variance of a given population is equal to some given variance, which we can call 02 . That is our null hypothesis is H0202: and our alternative hypothesis is H1202: . If the underlying data is normally distributed, we use the test ratio 20221sn , which has the chi-squared distribution with n 1 degrees of freedom (often written 12 n). For a two-tailed test we would pick two values of chi-squared, 1-2 and 222 , and accept the null hypothesis if 2 lies between them.For example, assume that we believe that the distribution of the ages of a group of workers is normal, and we wish to test our belief that the variance is 64. Our data is a sample of 17 workers, and our computations give us a sample variance of 100. Let us set our significance level at 2% and state our problem as follows: HH02126464:: n DF n s 17 1 16 100 64 02202, , , , . We can compute .25641001612022sn Since 201 16 . , and DF we go to the chi-squared table to find 000.32 and 812.5 16201.162.99. The 'accept' region is between these two values, so we cannot reject the null hypothesis.As a second example, assume that we are testing the same null hypothesis, but the sample size is 73, so that chi-squared has 72 degrees of freedom. Thus we have the following:n DF n s 73 1 72 100 64 02202, , , , . . The formula for the chi-squared statistic gives us .5.112641007212022sn If we cannot find an appropriate 2 on our table because of the high number of degrees of freedom, we use the z formula suggested in section 2. This is
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