1.) Given that Z is a standard normal random variable, find the following. Hint:

Question

1.) Given that Z is a standard normal random variable, find the following. Hint: draw a picture.

(a) P (Z > 0.47)

(b) P (-1.54 < Z < 0.51)

(c) The value z such that P (Z > z) = 0.0721

(d) The two z values (plus or minus z) such that P (-z < Z < z) = 0.9500

(e) The value z such that P (0 < Z < z) = 0.2995

2.) Suppose that the completion times for the Horsetooth Half marathon approximately follow a normal distribution with a population mean of 130 minutes with a population standard deviation of 20 minutes.

(a) What is the probability that a randomly selected runner of the Horsetooth half will take longer than 140 minutes to complete the race?

(b) Suppose that 500 runners complete the Horsetooth half marathon. How many of these runners are expected to finish in more than 140 minutes? (Hint: use part (a))

3.) Membership in Mensa requires an IQ score in the upper two percent of the general population (mensa.rog). Given that IQ scores for the general population have a normal distribution with a mean of 100 and a standard deviation of 15, what is the IQ score required to qualify for Mensa?

PLEASE SHOW WORK

Explanation / Answer

2)

mean = 130 , s = 20

a) P(x >140)
z = ( x - mean) / s
= ( 140 - 130) / 20
= 0.5
P(x >140) = P(z > 0.5) = 0.3085 (by using standard table)

b) n =500
P(x >140) = P(z > 0.5) = 0.3085 (by using standard table)
Hence, expected runners to finish = 500 * 0.3085
= 154.25

c)
z value at top 2% = 2.054
mean = 100 , s = 15
z = ( x - mean) / s
2.054 = ( x - 100)/ 15
x = 130.81

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