A random sample of 30 plants has a sample standard deviation of 3.4 days for maturity. A 95% confidence interval for the population variance is __ to __days.
To calculate the confidence interval for the population variance, we will use the Chi-square distribution.
Step 1: Determine the degrees of freedom.
In this case, the degrees of freedom are calculated as (n - 1), where n is the sample size. So, with a sample size of 30, the degrees of freedom will be 29.
Step 2: Find the critical values from the Chi-square distribution.
Since we want a 95% confidence interval, we need to find the critical values that leave a 2.5% tail on each side of the distribution. From the Chi-square distribution table or calculator, the critical values for the 2.5% tail for 29 degrees of freedom are approximately 15.09 and 45.72.
Step 3: Calculate the confidence interval.
The confidence interval for the population variance is calculated using the formula:
(((n - 1) * sample standard deviation^2) / Chi-square upper value, ((n - 1) * sample standard deviation^2) / Chi-square lower value)
Using the given sample standard deviation of 3.4 days, we can substitute the values into the formula:
(((30 - 1) * 3.4^2) / 45.72, ((30 - 1) * 3.4^2) / 15.09)
Simplifying the calculation, we get:
(29 * 3.4^2) / 45.72, (29 * 3.4^2) / 15.09
After evaluating the expression, the confidence interval for the population variance is approximately 1.561 to 3.913 days squared.