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
2.2< o < 14.3
Using a formula for confidence interval of variance with your data included, we have:
[(6-1)16.8]/(chi-square) to [(6-1)16.8]/(chi-square)
Use values from a chi-square table that equate to a 99% interval using 5 degrees of freedom, then finish the calculation for the interval.
To find the 99% confidence interval of the true variance, we can use the chi-square distribution.
Step 1: Define the degrees of freedom.
Since we have a sample size of 6 years, the degrees of freedom are determined by n-1, where n is the number of years. So, in this case, the degrees of freedom (df) are 6-1 = 5.
Step 2: Determine the critical values.
To find the critical values from the chi-square distribution, we need three pieces of information: the significance level (1 - confidence level), the degrees of freedom, and whether the confidence interval is one-tailed or two-tailed.
For a one-tailed 99% confidence interval, the significance level is 0.01 (1 - 0.99), and we have 5 degrees of freedom. Looking up the critical value in a chi-square distribution table or using statistical software, we find that the critical value is approximately 11.07.
Step 3: Calculate the confidence interval.
The confidence interval formula for the variance is (df * sample variance) / chi-square critical value.
Thus, the confidence interval for the true variance is:
(5 * 16.8) / 11.07 = 7.560.
Therefore, the 99% confidence interval of the true variance is (0, 7.560).
To find the 99% confidence interval of the true variance, we can use the chi-square distribution.
The chi-square distribution is used to estimate the true variance of a population based on a sample. The formula to calculate the confidence interval of the variance is as follows:
(Variance / (Chi-square Upper Value), Variance / (Chi-square Lower Value))
In this case, we have a sample size of 6 years and a variance of 16.8 deaths per year. However, since we are dealing with a chi-square distribution, we need to calculate the chi-square upper and lower values based on the sample size and the desired confidence level.
To find the chi-square values, we need to determine the degrees of freedom. In this case, the degrees of freedom is equal to the sample size minus 1, which is 6 - 1 = 5.
Using a chi-square distribution table or a calculator, we can find the chi-square values associated with a 99% confidence level and 5 degrees of freedom. In this case, the chi-square upper value is 15.086 and the chi-square lower value is 0.831.
Now, we can substitute these values into the formula to find the confidence interval of the true variance:
(16.8 / 15.086, 16.8 / 0.831)
(1.116, 20.234)
Therefore, the 99% confidence interval of the true variance is (1.116, 20.234) deaths per year.