Question

The average retail price of Gasoline(all types) for the first half of 2005 was 212.2 cents. What would the standard deviation have to be in order for a 24% probability that a gallon of gas costs less than $1.80? Round z-value calculations to 2 decimal places and final answer to the nearest cent

Answer 1

Given that the average retail price of Gasoline(all types) for the first half of 2005 was 212.2 cents.

Now here we need to find the the standard deviation in order for a 24%=24/100=0.24 probability that a gallon of gasoline costs less than $1.80=180 cents.

Let X=Price of a gallon of gasoline.

and,

Before we go on to find (*) let us know a bit about Normal Distribution and its properties.

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

A Property of Normal Distribution

If X~Normal(μ,σ²)

[By transformation of Random Variables]

Property of ?(x)

Since normal distribution is a symmetric distribution

Coming back to our problem,

[From property of Normal Distribution]

[Since X~Normal(μ=212.2,σ²)]

Now by property of ?(x),

Looking at the Biometrika Tables we see that ?(0.71)=0.76

Hence the standard deviation is 45 cents.