Question

9. Let X ~ N(186; 16). Find:

(a) P(X <= 215)

(b) P(140 < X < 197)

(c) The first quartile for X

(d) The third quartile for X

(e) the IQR for X

(f) P(|X-186|> 35)

10. A soft drink machine discharges an average of 325 ml per cup. The amount of drink is normally distributed with standard deviation of 21 ml. What fraction of cups will contain more than 366 ml? (Keep 4 decimals)

Answer 1

9)

Answer)

As the data is normally distributed we can use standard normal z table to estimate the answers

Z = (x-mean)/s.d

Given mean = 186

S.d = 16

A)

P(x<215)

Z = (215 - 186)/16

Z = 1.81

From z table, P(z<1.81) = 0.9649

B)

P(140<x<197) = P(x<197) - P(x<140)

P(x<197)

Z = (197 - 186)/16

Z = 0.69

From z table p(z<0.69) = 0.7549

P(x<140)

Z = (140 - 186)/16 = -2.88

From z table, P(z<-2.88) = 0.002

Required probability is 0.7549 - 0.002 = 0.7529

C)

Below first quartile, 25% of data lies

From z table, P(z<-0.67) = 25%

-0.67 = (Q1 - 186)/16

Q1 = 175.28

D)

Above third quartile, 25% lies

From z table, P(z>0.67) = 25%

0.67 = (Q3 - 186)/16

Q3 = 196.72

E)

IQR = Q3 - Q1 = 21.44