Question

the average

Answer 1

Hint:
Convert data into z-score.
\(\Large Z=\frac{X-\mu}{\sigma}\)
where
\(\sigma=\)standard deviation and
\(\mu=\)mean
Use the normal distribution table to find the probability corresponding to the calcualted z-score.
The following link might help too:
http://www.statisticshowto.com/how-to-calculate-a-z-score/

Answer 2

thank you..The middle being .95 means the outer two are .025. So i get 0.975 for the right side and tried to look up on the normal distribution table. which gave me 0.8365

Answer 3

but that is not correct

Answer 4

The question asks the minimum and maximum life expectancies.
You would have to convert the corresponding z-scores back into life-expectancies, namely in number of months.
Also, you'd be looking for z-scores for probability equaling .975 and .025

Answer 5

You can check the duplicate of your question for a different (easier?) method, but if you'd like to see what mathmate had in mind:

You're looking for \(a\) and \(b\) such that
\[\mathbb P(a\le X\le b)\]where \(X\) is a random variable representing the lifespan of a dishwasher. Since \(X\) is normally distributed, you can transform it to the standard normal distribution via the transformation in mathmate's post:
\[\mathbb P\left(\frac{a-6}{8}\le Z\le \frac{b-6}{8}\right)\]and since the distribution is continuous, you can split this up into two left-tailed probabilities
\[\mathbb P\left(Z\le\frac{b-6}{8}\right)-\mathbb P\left(Z\le\frac{a-6}{8}\right)\]Because the distribution is symmetric, and because you want this probability to be equal to \(0.95\), you want the above to yield
\[\color{red}{\mathbb P\left(Z\le\frac{b-6}{8}\right)}-\color{blue}{\mathbb P\left(Z\le\frac{a-6}{8}\right)}=\color{red}{0.975}-\color{blue}{0.025}=0.95\]Then using a table, e.g. the one here: http://dsearls.org/courses/M120Concepts/ClassNotes/Statistics/520A_LeftTailTable.htm
you can find the red probability to correspond to a \(z\) score of about \(1.9\), while the blue corresponds to \(-1.9\). (This would be more accurate than the empirical rule which suggests they would be \(2\) and \(-2\), respectively.)

From here, you would undo the transformation on the \(z\) scores to find the corresponding endpoints in the distribution for \(X\).

Answer 6

thank you for your help. how did you get 1.9 for 0.975 on table. i'm getting about 0.8340

Answer 7

Looks like you're doing the look-up backwards. The row and column headers are the actual \(z\) score, while the decimals in the grid are the probabilities.