Continuous Probability Distributions (Standard Normal Distribution) question: If the daily consumption of the average person follows a normal distribution that consists of the following..

Question

If the daily consumption of the average person follows a normal distribution that consists of the following:

3.58 (mean), 1.04 litres (standard deviation)

How many litres do 67.72 percent of consumers drink per day?

Answer 1

The question is , as is common, poorly written. No where in the question was it stated whether the condition was supposed to be less than, greater than, or not equal to. The previous answer assumed to test for a number of litters that 67.72% of the population drank equal to or LESS then, which makes little sense.

Taking the question literally,it is asking EXACTLY how many liters ANY specific amount of the population dranks. This is impossible to answer as we have assumed a continuous distribution and the probability of any specific number of liters being consumed is infinitesimally small. .

It is most reasonable to assume that the question means "What is the minimum number (x) of liters that 67.72% of the population would consume daily" Which means that everyone in that 67.72% sub population would have consumed at least x liters, as well as some unspecified amount more.

This is a right tailed distribution. We want to know what number x , given ℵ(μ,σ²). where mu = 3.58 and sigma squared = 1.0816, will satisfy the equation P(x>=μ) = .6772

This can be determined from tables, on line calculators, or hand calculators.

Using a TI-86 ( which gives the inverse normal distribution for the left sided distribution so we adjust the input to the function invnm())

we find that invnm((1-.6772),3.58,1.04) = 3.101721 liters.

or rounded, 3.1 l

Answer 2

x = liters drank

Find z-score associated with 67.72 percentile from standard normal probability table.. 0.460

z = (x - mean)/SD is normally distributed mean 0, SD 1

substitute z, mean and SD into equation to determine x at 67.72 percentile:

0.460 = (x - 3.58)/1.04

x = 1.04 * 0.460 + 3.58 = 4.06