Tough standard deviation problem

MubbashirAbbas wrote:A battery manufacturer claims that the mean life of its batteries is 5 yrs with the standard deviation of 10 months. If a particular battery chosen at random lasted only 37 months, what is the chance that the manufacturer's claims are false, based on the evidence, as well as on the fact that battery lives are normally distributed

A) 2.5% b) 97.5% c)4% d)5%

Please explain the steps to solve this problem. Thanks

Hi, there. I'm happy to contribute on this problem.

First of all, let me say emphatically: this is NOT a GMAT problem. The GMAT doesn't come anywhere close to this. This would be a typically problem in, for example, the AP Statistics course.

Keep in mind: essentially nothing of what I say here will be stuff you need to know for the GMAT. Nevertheless, I explain the solution.

Standard deviation is a measure of spread for any distribution, and it has an intimate relationship with the normal distribution (the Bell Curve). It is a kind of "yardstick" for the Bell Curve. On any normal distribution in the world, the following relationships are true:
68% of the data is within ±1 standard deviation of the mean
95% of the data is within ±2 standard deviations of the mean
99.7% of the data is within ±3 standard deviations of the mean

Here, the battery is suppose to last a mean of 5 yrs = 60 months. It dies after only 37 months, falling 23 months short of the mean.

Well, standard deviation = 10 months, so this means: 68% of batteries will last between 50 months and 70 months (i.e. within one standard deviation of the mean)
95% of batteries will last between 40 months and 80 months (i.e. within two standard deviation of the mean)
Think about that last scenario. Imagine a Bell Curve with vertical lines drawn, one two standard deviations above the mean, and the other two standard deviations below the mean. See the attached image. 95% of batteries are between those lines, and the 5% not between are split symmetrically between the two tails. 2.5% of batteries have a life shorter than 40 months, and 2.5% have a life longer than 80 months.

This battery has a life of 37 months, shorter than 40 months, and only 2.5% of batteries are within that range. Of the answers listed, A is the closest.

Technically, if we were going to calculate an exact answer, we would calculate the z-score (z = -2.3), which would give us a probability of P = 0.0107240881 --- that's the answer from a TI-84, a calculator with a full statistical package of software on it. Again, this would be a standard AP Statistics question, but it is way beyond anything on the GMAT.

Does all that make sense? Let me know if you have any further questions.

Mike