Question

Assume a normal distribution and find the following probabilities. (Round the values of z to 2 decimal places. Round your answers to 4 decimal places.)

(a) P(x < 18 | μ = 21 and σ = 4)

(b) P(x ≥ 43 | μ = 30 and σ = 8)

(c) P(x > 28 | μ = 30 and σ = 6)

(d) P(13 < x < 21 | μ = 19 and σ = 4)

(e) P(x ≥ 75 | μ = 60 and σ = 2.83)

Answer 1

Solution :

Given that ,

(a)

mean = = 21

standard deviation = = 4

P(x < 18) = P[(x - ) / < (18 - 21) / 4]

= P(z < -0.75)

= 0.2266

P(x < 18) = 0.2266

(b)

mean = = 30

standard deviation = = 8

P(x 43) = 1 - P(x   43)

= 1 - P[(x - ) / (43 - 30) / 8]

= 1 -  P(z 1.63)   

= 1 - 0.9484

= 0.0516

P(x 43) = 0.0216

(c)

mean = = 30

standard deviation = = 6

P(x > 28) = 1 - P(x < 28)

= 1 - P[(x - ) / < (28 - 30) / 6]

= 1 - P(z < -0.33)

= 1 - 0.3707

= 0.6293

P(x > 28) = 0.6293

(d)

mean = = 19

standard deviation = = 4

P(13 < x < 21) = P[(13 - 19)/ 4) < (x - ) /  < (21 - 19) / 4) ]

= P(-1.5 < z < 0.5)

= P(z < 0.5) - P(z < -1.5)

= 0.6915 - 0.0668

= 0.6247

P(13 < x < 21) = 0.6247

(e)

mean = = 60

standard deviation = = 2.83

P(x 75) = 1 - P(x   75)

= 1 - P[(x - ) / (75 - 60) / 2.83]

= 1 -  P(z 5.30)

= 1 - 1

= 0

P(x 75) = 0