. An automobile is selling a new model car with an advertised in‐city mileage of 27 MPG. As with most
car models, there is no associated measure of variability (variance, SD, etc.) included with the advertised
MPG. You write the manufacturer and request details on variability in MPG, and you find that the SD is 3
MPG. You assume that MPG for this model follows a Standard Normal Distribution.
a) If you were to buy one of these new cars, what is the probability that you would purchase one that
gets <20 MPG in‐city?
b) What is the probability that you would purchase one that gets >30 MPG i
To calculate the probabilities in this scenario, we can use the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. In this case, we have the mean (advertised MPG) and standard deviation (SD).
a) To calculate the probability that you would purchase a car that gets less than 20 MPG in-city, we need to calculate the z-score and then look up the corresponding probability in the standard normal distribution.
The z-score formula is:
z = (x - μ) / σ
Where:
x = 20 MPG (value we want to find the probability for)
μ = 27 MPG (mean)
σ = 3 MPG (standard deviation)
Plugging in the values, we get:
z = (20 - 27) / 3 = -7 / 3 = -2.3333
Now, we can look up the probability of a z-score of -2.3333 in the standard normal distribution table. The probability corresponds to the area under the curve to the left of the z-score.
Using a standard normal distribution table or a calculator, the probability is approximately 0.0099, or 0.99%.
Therefore, there is about a 0.99% chance that you would purchase a car that gets less than 20 MPG in-city.
b) To calculate the probability that you would purchase a car that gets more than 30 MPG in-city, we follow a similar process.
The z-score formula remains the same:
z = (x - μ) / σ
Where:
x = 30 MPG (value we want to find the probability for)
μ = 27 MPG (mean)
σ = 3 MPG (standard deviation)
Plugging in the values, we get:
z = (30 - 27) / 3 = 3 / 3 = 1
Now, we can look up the probability of a z-score of 1 in the standard normal distribution table. The probability corresponds to the area under the curve to the right of the z-score.
Using a standard normal distribution table or a calculator, the probability is approximately 0.1587, or 15.87%.
Therefore, there is about a 15.87% chance that you would purchase a car that gets more than 30 MPG in-city.