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 9 months. Other battery models, produced by similar processes, have normally distributed life spans. The 90% confidence interval for the population mean life of the new model is:

Answer 1

The 90% confidence interval would be given by (70.557;79.443)

Explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X=75` represent the sample mean

`\mu` population mean (variable of interest)

s=9 represent the sample standard deviation

n=13 represent the sample size

Calculate the confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=13-1=12`

Since the confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,12)".And we see that
`t_(\alpha/2)=1.78`.

Now we have everything in order to replace into formula (1):

`75-1.78(9)/(√(13))=70.557`

`75+1.78(9)/(√(13))=79.443`

So on this case the 90% confidence interval would be given by (70.557;79.443)