Question

The average gas mileage of a certain model car is 28 miles per gallon. If the gas

mileages are normally distributed with a standard deviation of 1.7, find the probability
that a car has a gas mileage of between 27.8 and 28.3 miles per gallon.

Answer 1

To find the probability of a car having a gas mileage between 27.8 and 28.3 miles per gallon, we can use the properties of a normally distributed random variable. By standardizing the values and using the standard normal distribution table or a calculator, we find that the probability is approximately 0.0241 or 2.41%.

Step-by-step explanation:

To find the probability that a car has a gas mileage between 27.8 and 28.3 miles per gallon, we can use the properties of a normally distributed random variable. First, we need to standardize the values using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

For 27.8 miles per gallon:

Z = (27.8 - 28) / 1.7 = -0.1176

For 28.3 miles per gallon:

Z = (28.3 - 28) / 1.7 = 0.1765

Next, we can use a standard normal distribution table or a calculator to find the cumulative probability corresponding to these Z-values. The probability can be calculated as P(27.8 ≤ X ≤ 28.3) = P(Z ≤ 0.1765) - P(Z ≤ -0.1176).

By looking up the probabilities in the standard normal distribution table or using a calculator, the probability is approximately 0.5706 - 0.5465 = 0.0241 (or 2.41%).

Answer 2

Answer: The probability is 0.3432.

Step-by-step explanation:

mu = 28, sigma = 0.45, x1 = 27.8, x2 = 28.

z1 = (x1 - mu)/sigma

z1 = (27.8 - 28)/0.45 = -0.4444

z2 = (x2 - mu)/sigma = (28.2 - 28)/0.45 = 0.4444

P(27.8 < x < 28.2) = P(-0.4444 < z < 0.4444) = 0.3432