Question

A normal random variable x has mean ? = 1.6 and standard deviation ? = 0.17. Find the probability associated with each of the following intervals. (Round your answers to four decimal places.)

(a)    

1.00 < x < 1.20

(b)    

x > 1.37

(c)    

1.35 < x < 1.50

Answer 1

Solution :

Given that ,

mean = = 1.6

standard deviation = = 0.17

(a)

P(1.00 < x < 1.20) = P((1.00 - 1.6 / 0.17) < (x - ) / < (1.20 - 1.6) / 0.17) )

P(1.00 < x < 1.20) = P(-3.53 < z < -2.35)

P(1.00 < x < 1.20) = P(z < -2.35) - P(z < -3.53) = 0.0094 - 0.0002 = 0.0092

Probability = 0.0092

(b)

P(x > 1.37) = 1 - P(x < 1.37)

= 1 - P((x - ) / < (1.37 - 1.6) / 0.17)

= 1 - P(z < -1.35)

= 1 - 0.0885

= 0.9115

P(x > 1.37) = 0.9115

Probability = 0.9115

(c)

P(1.35 < x < 1.50) = P((1.35 - 1.6 / 0.17) < (x - ) / < (1.50 - 1.6) / 0.17) )

P(1.35 < x < 1.50) =  P(-1.47 < z < -0.59)

P(1.35 < x < 1.50) =  P(z < -0.59) - P(z < -1.47) = 0.2776 - 0.0708 = 0.2068

Probability = 0.2068