Question

Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 80 people. (You may need to use the standard normal distribution table. Round your answers to the nearest whole number.)

(a) How many would you expect to be between 170 and 175 cm tall?


(b) How many would you expect to be taller than 177 cm?

Answer 1

Solution:

We are given

µ = 170

σ = 5

n = 80

Z = (Xbar - µ)/[σ/sqrt(n)]

Part a

Here, we have to find P(170<Xbar<175)

P(170<Xbar<175) = P(Xbar<175) – P(Xbar<170)

Find P(Xbar<175)

Z = (175 – 170)/[5/sqrt(80)]

Z = 5/ 0.559017

Z = 3.577709

P(Z< 3.577709) = P(Xbar<175) = 0.999827

(by using z-table or excel)

Now find P(Xbar<170)

Z = (170 – 170)/[5/sqrt(80)]

Z = 0

P(Z<0) = P(Xbar<175) = 0.5

(by using z-table)

P(170<Xbar<175) = P(Xbar<175) – P(Xbar<170)

P(170<Xbar<175) = 0.999827 – 0.5

P(170<Xbar<175) = 0.499827

Required probability = 0.499827

Required number = 80*0.499827 = 39.98614

Answer: 40

Part b

Here, we have to find P(Xbar>177) = 1 – P(Xbar<170)

Z = (177 - 170)/[5/sqrt(80)]

Z = 7/0.559017

Z = 12.52198

P(Z<12.52198) = P(Xbar<177) = 1

(by using z-table or excel)

P(Xbar>177) = 1 – P(Xbar<170)

P(Xbar>177) = 1 – 1

P(Xbar>177) = 0

Required number = 80*0 = 0

Answer: 0