Answer:
The lower limit of the 90% confidence interval for the population mean life of the new model is 72.53 months.
Step-by-step 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](https://tex.z-dn.net/?f=df%20%3D%2013-1%20%3D%2012)
Then, we need to subtract one by the confidence level
and divide by 2. So:
![\frac{1-0.90}{2} = \frac{0.10}{2} = 0.05](https://tex.z-dn.net/?f=%5Cfrac%7B1-0.90%7D%7B2%7D%20%3D%20%5Cfrac%7B0.10%7D%7B2%7D%20%3D%200.05)
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
.
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 = \frac{5}{\sqrt{13}} = 1.3868](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B5%7D%7B%5Csqrt%7B13%7D%7D%20%3D%201.3868)
Now, we multiply T and s
![M = T*s = 1.782*1.3868 = 2.47](https://tex.z-dn.net/?f=M%20%3D%20T%2As%20%3D%201.782%2A1.3868%20%3D%202.47)
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.