According to AAA, the price of a gallon of regular, unleaded gas across gas stations in North Carolina is normally distributed with a mean of $2.39 and a standard deviation of $0.15. Find the price such that the probability that a randomly chosen gas station charges more than that price is 20%.

The price at which the probability that a randomly chosen gas station charges more than that price is 20% is $2.52.

Step-by-step explanation:

We are given that the price of a gallon of regular, unleaded gas across gas stations in North Carolina is normally distributed with a mean of $2.39 and a standard deviation of $0.15.

Let X = price of a gallon of regular, unleaded gas across gas stations in North Carolina.

SO, X ~ Normal()

The z score probability distribution for the normal distribution is given by;

                                     Z  =    ~ N(0,1)

where, = population mean = $2.39

            = stnadard deviation = $0.15    

Now, we have to find the price such that the probability that a randomly chosen gas station charges more than that price is 20%, that means;

           P(X > x) = 0.20       {where x is the required price}

           P( > ) = 0.20

           P(Z > ) = 0.20

Now in the z table, the critical value of x which represents the top 20% area is given as 0.8416, that is;

                          x  =  2.39 + 0.13 = $2.52

Hence, the price at which the probability that a randomly chosen gas station charges more than that price is 20% is $2.52.

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