The lifetimes of all the batteries produced by a certain company in a year have a distribution that is symmetric about the mean m. If the distribution has a standard deviation of d, what percent of the distribution is greater than m + d ?

(1) 68 percent of the distribution lies in the interval from m - d to m + d, inclusive.

(2) 16 percent of the distribution is less than m - d.


Cam somebody explain? Thanks

Hi there,

The answer should be Either statement is sufficient!!

Note, the question is "What is the percentage of batteries outside m+d". What you need to understand is a distribution that is "symmetric about the mean". This means that is you draw a line at the mean, the distribution is identical on both sides. Hence, the number of items at a certain distance d [called standard deviation here] is going to the same on either side of the mean. Hence, to find the items which are beyond m+d, you need to know one of the following.

1. How many items[%age] are before (m-d) as percentage of items before (m-d) is the same as percentage of items after (m+d)
OR
2. What percentage of items are in the interval (m-d,m+d) as the number of items beyond (m+d) is 1/2*[100 - percentage of items between interval (m-d, m+d).

I hope that the answer helps

the poster above did a good job of explaining the significance of the symmetry statement, so i don't need to rehash that.
thanks, poster above.

what the poster above didn't write is that it actually makes no difference at all that 'd' is the standard deviation in this problem. 'd' could be a completely random number and the problem would still work out the same way.
they're writing the problem with 'd' as the standard deviation for at least one, and probably both, of the following reasons: (a) to mess with your head, and (b) because IF 'd' IS the standard deviation, then the 68% and 16% take on a special significance (which is irrelevant here, but of tremendous importance in statistics).

but:

just think about SYMMETRY.

think of 'm' in the middle of a number line. then 'm + d' is just as far to the right of it as 'm - d' is to the left of it.

so:

(1) 68% of the stuff lies between m - d and m + d.
this means the other 32% (= 100% - 68%) of the stuff lies outside those boundaries.
because of the symmetry in the problem, this means that 16% of the stuff is to the left of m - d, and 16% of the stuff is to the right of m + d.
answer = 16%
sufficient

(2) by symmetry, the amount of stuff to the left of m - d must be the same as the amount of stuff to the right of m + d (because those two regions are mirror images of each other under the symmetry in the problem).
thus, answer = 16%
sufficient

answer = d