Question

1. A sample size of 80 will be drawn from a population with mean 20 and standard deviation 15. Find the probability that x̄ will be between 18 and 21.

A) 0.6087 B) 0.5841 C) 0.1170 D) 0.2743

2. A certain car model has a mean gas mileage of 29 miles per gallon (mpg) with a standard deviation 5 mpg. A pizza delivery company buys 47 of these cars. What is the probability that the average mileage of the fleet is greater than 27.9 and 29.3 mpg?

A) 0.5936 B) 0.0655 C) 0.6591 D) 0.4064

Answer 1

#1.
Here, μ = 20, σ = 15/sqrt(80) = 1.6771, x1 = 18 and x2 = 21. We need to compute P(18<= X <= 21). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (18 - 20)/1.6771 = -1.19
z2 = (21 - 20)/1.6771 = 0.6

Therefore, we get
P(18 <= X <= 21) = P((21 - 20)/1.6771) <= z <= (21 - 20)/1.6771)
= P(-1.19 <= z <= 0.6) = P(z <= 0.6) - P(z <= -1.19)
= 0.7257 - 0.117
= 0.6087

Option A

#2.
P(27.9 <= X <= 29.3) = P((29.3 - 29)/0.7293) <= z <= (29.3 - 29)/0.7293)
= P(-1.51 <= z <= 0.41) = P(z <= 0.41) - P(z <= -1.51)
= 0.6591 - 0.0655
= 0.5936

option A