A study of generator related carbon monoxide deaths showed that a sample of 6 recent years had a variance of 16.8 deaths per year. Find the 99% confidence interval of the true variance
To find the 99% confidence interval of the true variance, we can use the chi-square distribution. The chi-square distribution is commonly used for inference about the variance in statistics.
Let's follow these steps to calculate the confidence interval:
Step 1: Determine the degrees of freedom.
The degrees of freedom (df) for variance is calculated as n - 1, where n is the sample size. In this case, the sample size is 6, so the degrees of freedom would be 6 - 1 = 5.
Step 2: Find the critical values.
Since we want a 99% confidence interval, we need to find the critical values for the chi-square distribution. These critical values depend on the degrees of freedom and the desired confidence level.
Using a chi-square distribution table or statistical software, we can find the critical values associated with a confidence level of 99% and degrees of freedom of 5. The critical values are the values that bound the confidence interval.
In this case, the critical values are approximately 2.571 (lower bound) and 14.067 (upper bound).
Step 3: Calculate the confidence interval.
The confidence interval for the true variance can be calculated using the formula:
[ (n - 1) * s^2 / chi-square upper bound, (n - 1) * s^2 / chi-square lower bound ]
where n is the sample size, s^2 is the sample variance, and chi-square upper bound/lower bound are the critical values from Step 2.
Plugging in the given values, we get:
[ (6 - 1) * 16.8 / 14.067, (6 - 1) * 16.8 / 2.571 ]
Simplifying, we get:
[ 84 / 14.067, 84 / 2.571 ]
The confidence interval for the true variance is approximately:
[ 5.992, 32.697 ]