8.6:
Finding Critical Values for Chi-Square
Consider sample data on the fuel economy of some car brands. To obtain a 95% confidence interval for the population standard deviation, one must calculate the critical values that separate the likely results from the unlikely ones.
A 95% confidence level covers 95% of the area under the curve, while the remaining 5% area distributed equally on either side.
As the chi-square distribution is asymmetrical, the right and left critical values separating an area of 2.5% or 0.025 on both sides are individually determined.
To determine the right-tailed critical value, locate nine on the left column for degrees of freedom in the chi-square table and find 0.025 across the top row, yielding a value of 19.023.
Since the table provides cumulative areas to the right of the critical value, subtract the remaining 0.025 area from the total area under the curve to obtain 0.975. Now, using the chi-square table, the left-tailed critical value is calculated.
8.6:
Finding Critical Values for Chi-Square
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.