Consider a curve representing sample data drawn randomly from a normally distributed population. One must construct confidence intervals to estimate or to test a claim regarding the population standard deviation. For example, a 95% confidence interval covers 95% of the area under the curve, and the remaining 5% is equally distributed on either side of the curve. To achieve such confidence intervals, one must determine the critical values. The critical values are simply the values separating the likely values from the unlikely ones.
As the chi-square distribution is asymmetrical, the left and right critical values separating an area of 2.5% or a significance level of 0.025 on either side of the curve are determined separately using tables. In the table for the chi-square critical values, critical values are found by first locating the row corresponding to the appropriate number of degrees of freedom df, where df = n - 1, n represents the sample size. The significance level α is used to determine the column. The right-tailed value is calculated by locating the area of 0.025 at the top of the table. Since the table is based on cumulative values from the right, for the left-tailed value, subtract 0.025 from the total area under the curve, that is, 1, and yields 0.975. The value in the corresponding column of 0.975 gives the left-tailed critical value.
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