Hypothesis Tests Concerning the Value of the Variance a Population - Investment Analysis
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The test statistic that is used to decide whether to reject the null or alternative hypothesis is the chi-square (χ2) statistic, which is defined as:
= (n-1)Sx2/
The chi-square distribution is a family of distribution each of which is defined by its degrees of freedom. The number of degrees of freedom is n-1.
It is to be noted that the chi square statistic is an adequate test statistic for testing hypotheses concerning population variances for normally distributed populations; it is not a good test statistic for testing hypotheses about the variances of populations that are not normally distributed.
Example: An investment management firm submits its investment record for the last 10 years. The std. Deviation of returns if 19%. But the firm claims that its targeted std. Deviation is 15%. Test at 5% significance level whether it is plausible.
Null Hypo: H0 = Sigma square of x is < 225
Alt. Hypo: H1 = Sigma square of x is > 225.
The degrees of freedom are 9, which is n-1 or 10-1. The calculated chi square value is
= (n-1)Sx2/sigma square = (1—1)(19)2/152
= 14.44
Null hypothesis is stated as one tailed test. Hence chi-square (.05, and 9) is 16.919. The calculated value of Chi square falls within the acceptance range of 0 to 16.919, thus the null hypothesis cannot be rejected.
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