Question

The engines made by Ford for speedboats had an average power of 220 horsepower (HP) and a standard deviation of 15 HP. The population distribution can be assumed to be normal.

a. Find the probability that an engine, chosen at random, will have a horsepower HP less than 215. Graph the situation. Shade in the area to be determined.

b. Find the standard error for a sample mean of 25 engines

c. A potential buyer intends to take 25 engines and will not place an order if their average horsepower is less than 215. Determine the probability that the buyer will not place an order? Graph the situation. Shade in the area to be determined.

Answer 1

Solution:
Given in the question
Mean () = 220
Standard deviation () = 15
population distribution can be assumed to be normal
Solution(a)
We need to calculate the probability that an engine, chosen at random, will have a horsepower HP less than 215, which can be calculated as
P(X<215) = ?, here we will standard normal distribution, First we will calculate Z-score which can be calculated as
Z-score = (X-)/ = (215-220)/15 = -5/15 = -0.33
From Z table, found a p-value
P(X<215) = 0.3707
So there is a 37.07% probability that t an engine, chosen at random, will have a horsepower HP less than 215.
Solution(b)
If Sample size n = 25
Than standard error of the sampling distribution = /sqrt(n) = 15/sqrt(25) = 3
Solution(c)
A potential buyer intends to take 25 engines and will not place an order if their average horsepower is less than 215. We required to determine the probability that the buyer will not place an order
i.e. P(X<215) =?
Z=(215-220)/3 = -1.67
From Z table we found p-value = 0.0475
P(X<215) = 0.0475
so there is 4.75% probability that the buyer will not place an order.