Find the critical values for a 98% confidence interval using the chi-square distribution with 20 degrees of freedom.
A) 7.633, 36.191
B) 7.434, 39.997
C) 8.260, 37.566
D) 9.237, 35.020

Question

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To find the critical values for a 98% confidence interval using the chi-square distribution with 20 degrees of freedom, we look at the chi-square distribution table. Specifically, we want to find the values that enclose 98% of the area in the right tail of the distribution.

Using the table, we find that the critical value for the right tail with 20 degrees of freedom and 0.01 level of significance (for a two-tailed test, which is equivalent to a 98% confidence interval) is 31.410. The critical value for the left tail is the same, but with a negative sign.

Therefore, the critical values for a 98% confidence interval using the chi-square distribution with 20 degrees of freedom are -31.410 and 31.410. However, since we are interested in the positive values, we take the positive of these critical values, which are 31.410 and 31.410.

Next, we need to find the values that correspond to the area between these critical values. We can do this by subtracting the cumulative probability at the lower critical value from the cumulative probability at the upper critical value, both calculated using the chi-square distribution with 20 degrees of freedom.

Using a chi-square distribution table or calculator, we find that the cumulative probability at 31.410 is approximately 0.990 and the cumulative probability at 8.260 is approximately 0.020. Therefore, the area between the critical values is approximately 0.990 - 0.020 = 0.970 or 97%.

Finally, we check the answer choices to see which one contains the critical values that enclose a 97% area in the right tail of the chi-square distribution with 20 degrees of freedom. Only choice C, 8.260 and 37.566, satisfies this condition.

Find the Critical Values for a 98% Confidence Interval Using