Standard Deviation?

Question

The average amount customers at a certain grocery store spend yearly is $636.55. Assume the variable is normally distributed. If the standard deviation is $89.46, find the probability that a randomly selected customer spends between $550.67 and $836.94.

Answer 1

First thing to do in this type of problem is normalize everything. That means subtract the mean and divide by the standard deviation, so that we're dealing with a standard normal distribution (mean = 0, standard deviation = 1).

So really, we're looking for the probability, in a standard normal distribution, that the variable is between the two bounds A and B:

A = (550.67 - 636.55) / 89.46 = -0.96

B = (836.94 - 636.55) / 89.46 =  2.24

Generally when you want to find the probability that a random variable falls between two values A and B, you use the cumulative distribution function (CDF) for that variable:

P(variable between A and B) = CDF(B) - CDF(A)

where CDF is the cumulative distribution function for that variable. In other words, you sum all the probability to the left of B, then subtract off all the probability to the left of A, and you're left with all the probability between A and B.

In the case of standard normal variables, the CDF can be looked up in a table phi(x), which I assume you have. The tables usually only define positive values, so we use the property of symmetry that says phi(-x) = 1-phi(x).

Now we can get the answer:

CDF(B) - CDF(A)

= phi(B) - phi(A)

= phi(2.24) - phi(-0.96)

= phi(2.24) - (1-phi(0.96))

By looking up the two values and plugging into the above, you'll get the desired probability.