Question

The fuel efficiency in miles per gallon of all bmw 320i's is (approximately) normally distributed with a mean of 25 and a standard deviation of 2. a dealer receives a shipment of a random sample of 320i's (random with respect to m.p.g., that is) from the factory. find the probability that average fuel efficiency for this shipment is less than 24 miles per gallon if the dealer receives (i) one car, (ii) four cars and (iii) sixteen cars. explain briefly why these three answers differ.

Answer 1

Given:
μ = 25 mpg, the population mean
σ = 2 mpg, the population standard deviation

If we select n samples for evaluation, we should calculate z-scores that are based on the standard error of the mean.
That is,
$z= \frac{x-\mu}{\sigma / \sqrt{n} }$

The random variable is x = 24 mpg.

Part (i):  n = 1
σ/√n = 2
z = (24 -25)/2 = -0.5
From standard tables,
P(x < 24) = 0.3085

Part (ii): n = 4
σ/√n = 1
z = (24 -25)/1 = -1
P(x < 24) = 0.1587

Part (iii): n=16
σ/√n = 0.5
z = (24 - 25)/0.5 = -2
P(x < 24) = 0.0228

Explanation:
The larger the sample size, the smaller the standard deviation.
Therefore when n increases, we are getting a result which is closer to that of the true mean.