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%.

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([tex]\mu=\$2.39,\sigma^{2} =\$0.15^{2}[/tex])

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

Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)

where, [tex]\mu[/tex] = population mean = $2.39

[tex]\sigma[/tex] = 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( [tex]\frac{X-\mu}{\sigma}[/tex] > [tex]\frac{x-2.39}{0.15}[/tex] ) = 0.20

P(Z > [tex]\frac{x-2.39}{0.15}[/tex] ) = 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;

[tex]\frac{x-2.39}{0.15}= 0.8416[/tex]

[tex]x-2.39} = 0.8416 \times 0.15[/tex]

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.