The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence to construct a confidence interval for the population variance and the population standard deviation of the reserve capacities of automotive batteries.

Question

17-03-2025
Mathematics

High School

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The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence to construct a confidence interval for the population variance and the population standard deviation of the reserve capacities of automotive batteries.

Answer :

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TomSmith675 TomSmith675 · Answer 1 · 2025-04-23T03:14:03-04:00

Final answer:

To find an 80% confidence interval for the population variance and standard deviation of automotive battery capacities, one uses the sample standard deviation and degrees of freedom in conjunction with a chi-square distribution table.

Explanation:

To construct an 80% confidence interval for the population variance and population standard deviation of the reserve capacities of automotive batteries, given a sample standard deviation of 0.25 hour from 18 samples, one can use the chi-square distribution.

To find the confidence interval for the population variance ($ ext ext ext ext ext), we need two chi-square values ($ ext ext ext ext ext) from the chi-square distribution table corresponding to the degrees of freedom ($ ext ext ext ext ext) which is n-1 (in this case, 17) and the desired confidence level (80%). The lower chi-square value corresponds to the higher percentile (60%) and the upper chi-square value corresponds to the lower percentile (20%), because of how the chi-square distribution is set up with respect to confidence intervals.

The formula to calculate the confidence interval for variance is given by:

CI for variance = $ ext ext ext ext ext

The formula for the standard deviation is just the square root of the variance, hence the CI for standard deviation can be obtained accordingly.