Standard Deviation problem (Big 5lb Book)

[maria08]

Hi

I've got the following problem from the big M book, chapter 22.

"The length of bolds made in factory Z is normally distributed with a mean length of 0.1630 meters and a SD of 0.0084 meters. Whats the probability that a randomly selected bold it between 0.1546 meters and 0.1756 meters long is between:

A. 54% and 61%
B. 61% and 68%
C. 68% and 75%
D. 75% and 82%
E. 82% and 89%

It soon became clear that the solution was lying between m-d and m+1.5d (m=mean and d= standard deviation).
To my knowledge, 0.5d in this case would be 13.5/2, that is 6.75.
Hence, the total was 34%+34%+6.75%= 74.75%.
I chose answer C.

Can someone advise where I went wrong.
Thanks in advance
L.

[tommywallach]

Hey Lore,

This is explained pretty well at the end of the chapter, but I'll try my best to add to it. Because of the shape of a bell curve, there is more "area" under the curve when it's closer to the mean (the high point). This is because the horizontal distance between SDs is constant. So if the hump is taller, there's more area inside it. This should be clear, because the area from mean to 1SD = 34% of the data, while the area from 1SD to 2SD is only 14%.

We know that 1SD = 34%

We know that the area from 1SD to 2SD = 14%

Now, because we know this data point is 1.5 SDs away, we have to work out how that .5 relates to 14%.

Half of 14% would be 7%. So will 1.5SDs have MORE than 7% or LESS than 7%? Because there's more area in the "left" half of the curve from 1SD to 1.5SD, there will be MORE than 7% of the data inside it.

34% + 34% + more than 7% = more than 75%, hence (D).

Hope that's a little clearer!

-t