Question

Could someone help me with this problem. I don't know how my prof got the answer.

Answer 1

where don't you understand?

Answer 2

umm why is it 95%, I don't know why its 2 standard deviations
away from the mean

Answer 3

@dan815

Answer 4

Say \(X\) is the time for a worker to complete the operation. \(X\) is normally distributed ("mound shaped"), so the problem asks for \(P(X>16.2)\).

You can standardize \(X\) to obtain \(Z\), a z-score, by subtracting off the mean and dividing by the standard deviation.

\[P(X>16.2) =P\left( \frac{X-12.8}{1.7}>\frac{16.2-12.8}{1.7}\right)=P(Z>2)\]

This is where the 2 standard deviations comes from. So they are looking for the proportion of people above 2 standard deviations. Either you can looking this up in a standard normal table, or recall that about 95% of the data are between -2 and 2 standard deviations from the mean, so the proportion above 2 standard deviations is half of 5% = 2.5% (the region beyond -2 and 2)