Answer:
The 90% confidence interval for the true mean lifetime of all batteries of this brand is between 39.3 hours and 41.7 hours. The lower limit is 39.3 hours while the upper limit is 41.7 hours.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 - 0.9}{2} = 0.05[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.05 = 0.95[/tex], so Z = 1.645.
Now, find the margin of error M as such
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 1.645\frac{9}{\sqrt{150}} = 1.2[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 40.5 - 1.2 = 39.3 hours
The upper end of the interval is the sample mean added to M. So it is 40.5 + 1.2 = 41.7 hours
The 90% confidence interval for the true mean lifetime of all batteries of this brand is between 39.3 hours and 41.7 hours. The lower limit is 39.3 hours while the upper limit is 41.7 hours.
