Question

1A. Let z denote a random variable having a normal distribution with μ = 0 and σ = 1. Determine each of the probabilities below. (Round all answers to four decimal places.)

(a) P(z < 0.1) =

(b) P(z < -0.1) =

(c) P(0.40 < z < 0.84) =

(d) P(-0.84 < z < -0.40) =

(e) P(-0.40 < z < 0.84) =

(f) P(z > -1.26) =

(g) P(z < -1.49 or z > 2.50) =

1B. Find the following probabilities for X = pulse rates of group of people, for which the mean is 76 and the standard deviation is 8. Assume a normal distribution. (Round all answers to four decimal places.)

(a) P(X ≤ 68).


(b) P(X ≥ 82).


(c) P(56 ≤ X ≤ 92).

Answer 1

Solution:

1 )Given that,

Using standard normal table,

a ) P ( Z < 0.1 )

= 0.5398

Probability = 0.5398

b ) P ( Z < -0.1 )

=0.4602

Probability = 0.4602

c ) P(0.40 < z < 0.84)

P ( Z < 0.84 ) - P ( Z < 0.40  )

= 0.7995 - 0.6554

= 0.1441

Probability = 0.1441

d ) P( -0.84 < z < -0.40)

P ( Z < -0.40 ) - P ( Z < -0.84 )

=0.3446 - 0.2005

= 0.1441

Probability = 0.1441

e ) P( -0.40 < z < 0.84)

P ( Z < 0.84 ) - P ( Z < -0.40  )

= 0.7995 - 0.3446

=0.4549

Probability = 0.4549

f ) P ( Z >-1.26 )

= 1 - P ( Z < -1.26)

= 1 - 0.1038

= 0.8962

Probability = 0.8962

g) P(z < -1.49 or z > 2.50)

P ( Z < -1.49 )

=0.0681

P ( Z > 2.50)

= 1 - P ( Z < 2.50)

= 1 - 0.9938

=0.0062

Probability =0.0681 +0.0062 =0.1043

2 )Given that,

mean = = 76

standard deviation = =8

a) P( x ≤ 68 )

P ( x - / ) ≤( 68 - 76 / 8)

P ( z ≤ - 8 / 8 )

P ( z ≤ -1 )

= 0.1587

Probability = 0.1587

b ) P (x > 82 )

= 1 - P (x ≤ 82)

= 1 - P ( x -  / ) ≤ ( 82 - 76 / 8)

= 1 - P ( z ≤ 6 / 8 )

= 1 - P ( z ≤ 0.75 )

Using z table

= 1 - 0.7734

= 0.2266

Probability = 0.2266

c ) P (56 ≤ x ≤ 92 )

P ( 56 - 76 / 8) < ( x -  / ) ≤ ( 92 - 76 / 8)

P ( - 20 / 8 ≤ z ≤ 16 / 8 )

P (-2.5 ≤ z ≤ 2.00 )

P ( z ≤ 2.00 ) - P ( z ≤ -2.5 )

Using z table

=0.9772 - 0.0062

= 0.9710

Probability = 0.9710