Probability

Question

The pharmaceutical company buys 400 batteries for running some specialist equipment. These batteries have a mean life of 155 hours and a standard deviation of 30 hours. Assume the battery life follows a normal distribution.

a) Use the standard normal table provided to calculate:
(i) the probability that a battery may work for less than 200 hours.
(ii) the probability that a battery may work for more than 200 hours.
(iii) the probability that a battery may work for less than 90 hours.

b) b) Find the number of batteries which have a life between 155 hours and 190 hours.

Answer 1

A Gaussian or normal distribution is common in statistics, and has well known formulas:

 

http://en.wikipedia.org/wiki/Normal_distribution

 

The standard normal table is derived from these formulas:

 

http://en.wikipedia.org/wiki/Standard_normal_table

 

We know that there are N=400 batteries in this sample...they have \mu=155 hours of average battery life, and have standard deviation \sigma=30 hours.

 

(i) We can now view the batteries that last more or less than 200 hours as lying some number of standard deviations away from the average...namely 200 = \mu + 1.5 * \sigma...this multiple of the standard deviations is the Z value in the table!  Be sure to remember that half of the batteries last for less than 155 hours, and calculate (using the table and z=1.5 standard deviations) the proportion of batteries between 155 and 200.

 

(ii) The probability of more than 200 hours, would be 1-prob(less than 200hrs).

 

(iii) For 90hrs we should again consider the number of standard deviations away from the average...(155-90)/30=z

This z value tells us how many are expected to be between 90 and 155...so 1/2 - this table value will tell us how many work for less than 90hours

 

For part b) we should redo parts (i) and (ii) for 190hours...then take 1/2 - Pr(battery lasts more than 190hrs) to find the probability a battery lasts between 155 and 190 hours...Multiply this expected proportion by the total number (400) to find the number with life between 155 and 190.

 

Hope that helps!