Question

Life tests performed on a sample of 13 batteries of a new model indicated: (1) an average life of 75 months, and (2) a standard deviation of 5 months. Other battery models, produced by similar processes, have normally distributed life spans. The lower limit of the 90% confidence interval for the population mean life of the new model is _________. (Specify your answer to the 2nd decimal.)

Answer 1

The lower limit of the 90% confidence interval for the population mean life of the new model is 72.53 months.

Explanation:

Our sample size is 13.

The first step to solve this problem is finding our degrees of freedom, that is, the sample size subtracted by 1. So

`df = 13-1 = 12`

Then, we need to subtract one by the confidence level
`\alpha` and divide by 2. So:

`(1-0.90)/(2) = (0.10)/(2) = 0.05`

Now, we need our answers from both steps above to find a value T in the t-distribution table. So, with 12 and 0.05 in the t-distribution table, we have
`T = 1.782`.

Now, we find the standard deviation of the sample. This is the division of the standard deviation by the square root of the sample size. So

`s = (5)/(√(13)) = 1.3868`

Now, we multiply T and s

`M = T*s = 1.782*1.3868 = 2.47`

The lower end of the interval is the mean subtracted by M. So it is 75 - 2.47 = 72.53

The lower limit of the 90% confidence interval for the population mean life of the new model is 72.53 months.