Question

Find the 99% confidence interval for the variance and standard deviation for the time it takes state police inspector to check a truck for safety if sample of 23 trucks has a standard deviation of 5.3 minutes. Assume the variable is normally distributed.

Answer 1

To calculate the 99% confidence interval for the variance and standard deviation, use the sample variance, sample size, degrees of freedom, and Chi-Square critical values. For variance, the formula includes the sample variance and degrees of freedom. The standard deviation interval is then the square root of the variance interval endpoints.

Step-by-step explanation:

To find the 99% confidence interval for the variance and standard deviation of the time it takes for a state police inspector to check a truck for safety when the sample size is 23 and the sample standard deviation is 5.3 minutes, and assuming the times are normally distributed, we use the Chi-Square distribution.

For a variance confidence interval, we use:

1. Sample variance (s2) = (Sample Standard Deviation)2 = 5.32 = 28.09
2. Sample size (n) = 23
3. Degrees of freedom (df) = n - 1 = 22
4. We then find the critical values from the Chi-Square distribution for df = 22 at the 99% confidence level. These are typically found in a Chi-Square table or using statistical software.
5. The formula for the confidence interval for the variance is given by:

Variance Confidence Interval =
(s2 × (n-1)) / χ(1-α/2;df), (s2 × (n-1)) / χ(α/2;df)

where:

• χ(1-α/2;df) and χ(α/2;df) are the Chi-Square critical values
• α = 1 - confidence level

To find the standard deviation confidence interval, simply take the square root of both ends of the variance confidence interval.