Assume each sample is taken from a normally distributed population and construct

Question

Assume each sample is taken from a normally distributed population and construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation.   A lawn mower manufacturer is trying to determine the standard deviation of the life of one of its lawn mower models. To do this, it randomly selects 12 lawn mowers that were sold several years ago and finds that the sample standard deviation is 3.25 years. Use a 99% level of confidence. Does this problem involve Chi distribution or test?

Explanation / Answer

Here sample standard deviation s= 3.25 years

So so sample variance s2 = 3.252 = 10.5625

This problem involves chi distribution

(a) 99% confidence interval for Population variance i s

(n-1)s2/X2/2,dF < 2 < (n-1)s2/X21- /2,dF

here n = 12 and = 0.01

X2/2,dF = X20.005,11 = 26.75685

X21-/2,dF =X20.995,11 = 2.603222

so Lower limit = (n-1)s2/X2/2 = (12-1) * 10.5625 /26.75685 = 4.3423

Upper limit = (n-1)s2/X21- /2,dF = (12-1) * 10.5625/ 2.603222 = 44.6322

so 99% confidence interval for variancve = (4.3423, 44.6322)

(ii) 99% level of confidence interval the population standard deviation =

sqrt [(n-1)s2/X2/2,dF]   < < sqrt [(n-1)s2/X21- /2,dF ]

sqrt (4.3423) < < sqrt(44.6322)

2.0838 < < 6.6807

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