Question

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%.

Answer 1

To find the price such that the probability that a randomly chosen gas station charges more than that price is 20%, we can use the z-score formula. The price is $2.516.

Step-by-step explanation:

To find the price such that the probability that a randomly chosen gas station charges more than that price is 20%, we can use the z-score formula. The z-score is calculated as (x - mean) / standard deviation, where x is the price we want to find. We need to find the z-score corresponding to a cumulative probability of 0.8 (1 - 0.2), which is 0.84. Rearranging the formula, we have x = (z-score * standard deviation) + mean. Substituting the values, we get x = (0.84 * 0.15) + 2.39 = 2.516.

Answer 2

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(
`\mu=\$2.39,\sigma^(2) =\$0.15^(2)`)

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

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean = $2.39

`\sigma` = 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(
`(X-\mu)/(\sigma)` >
`(x-2.39)/(0.15)` ) = 0.20

P(Z >
`(x-2.39)/(0.15)` ) = 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)/(0.15)= 0.8416`

`x-2.39} = 0.8416 * 0.15`

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.