Use technology to solve the following problem: A certain car model has a mean gas mileage of 33 miles per gallon (mpg) with a standard deviation 3 mpg. A pizza delivery company buys 48 of these Cars. What is the probability that the average mileage of the fleet is between 32.7 and 33.3 mpg?
A) 0.5116
B) 0.4884
C) 0.2442
D) 0.7558

Question

Quizplus · Accepted Answer

Using the Central Limit Theorem, we can calculate the probability that the sample mean falls within a certain range. The standard error (SE) of the mean is given by the formula SE = σ / √n, where σ is the standard deviation and n is the sample size. Here, σ = 3 mpg and n = 48 cars. Thus, SE = 3 / √48 ≈ 0.433. To find the probability that the average mileage is between 32.7 and 33.3 mpg, we convert these values to Z-scores: Z = (X - μ) / SE, where X is the value of interest, μ is the mean, and SE is the standard error. For 32.7 mpg, Z = (32.7 - 33) / 0.433 ≈ -0.693. For 33.3 mpg, Z = (33.3 - 33) / 0.433 ≈ 0.693. Using a standard normal distribution table or calculator, the probability between these Z-scores is approximately 0.5116.

Use Technology to Solve the Following Problem: a Certain Car