Statistics probability

Question

In a recently conducted study families reported using an average of 70 gallons of gas per month with a standard deviation of 9 gallons. If a family is chosen at random, what is the probability that it uses between 80 and 86 gallons of gas per month? Assume the data is normally distributed.

Answer must be in 4 digits

Answer 1

Here you have a random variable X which is normally distributed with N(70, 9). What is the probability if a family is chosen at random they would use between 80 and 86 gallons?

Well we need to normalize the random variable X (distributed normally) so we can calculate the probability named in the question. We use Xbar the average as an unbiased estimator of mu (the mean). The standard deviation is given.

Z = (X - Xbar)/Std

Thus P(80 < X < 86) = P ((80-70)/9 < Z < (86-70)/9) = P(1.11 < Z < 1.78)

You can look at the charts or use a TI-84 to find the area under the standard normal curve between z= 1.11 and z = 1.78

normalcdf(80,86,70,9) = .0955

Thus the probability of randomly chosen family using between 80 and 86 gallons of gas is .0955.