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 < 17 | μ = 20 and σ = 3)
enter the probability of fewer than 17 outcomes if the mean is 20 and the standard deviation is 3

(b) P(x ≥ 61 | μ = 50 and σ = 8)
enter the probability of 61 or more outcomes if the mean is 50 and the standard deviation is 8

(c) P(x > 45 | μ = 50 and σ = 5)
enter the probability of more than 45 outcomes if the mean is 50 and the standard deviation is 5

(d) P(16 < x < 19 | μ = 18 and σ = 3)
enter the probability of more than 16 and fewer than 19 outcomes if the mean is 18 and the standard deviation is 3

(e) P(x ≥ 75 | μ = 60 and σ = 2.79)
enter the probability of 75 or more outcomes if the mean is 60 and the standard deviation is 2.79

Answer 1

Solution :

a) Given, X follows Normal distribution with,

   = 20

   = 3

Find P(X < 17)

= P[(X - )/ <  (17 - )/]

= P[Z <  (17 - 20)/3]

= P[Z < -1.00]

= 0.1587 ... ( use z table)

P(X < 17) = 0.1587

b) Given, X follows Normal distribution with,

   = 50

   = 8

Find P(X > 61)

= P[(X - )/ >  (61 - )/]

= P[Z > (61 - 50)/8]

= P[Z > 1.38]

= 1 - P[Z < 1.38]

= 1 - 0.9162    ( use z table)

= 0.0838

P(X > 61) = 0.0838

c) Given, X follows Normal distribution with,

   = 50

   = 5

Find P(X > 45)

= P[(X - )/ >  (45 - )/]

= P[Z > (45 - 50)/5]

= P[Z > -1.00]

= 1 - P[Z < -1.00]

= 1 - 0.1587 ( use z table)

= 0.8413

P(X > 45) = 0.8413

d) Given, X follows Normal distribution with,

   = 18

   = 3

Find, P(16 < x< 19)

= P(X < 19) - P(X < 16)

=  P[(X - )/ <  (19 - 18)/3] -   P[(X - )/ <  (16 - 18)/3]

= P[Z < 0.33] - P[Z < -0.67]

= 0.6293 - 0.2514    ..Use z table

= 0.3779

P(16 < x< 19) = 0.3779

e) Given, X follows Normal distribution with,

   = 60

   = 2.79

Find P(X > 75)

= P[(X - )/ >  (75 - )/]

= P[Z > (75 - 60)/2.79]

= P[Z > 5.38]

= 1 - P[Z < 5.38]

= 1 - 1 ( use z table)

= 0.0000

P(X > 75) = 0.0000