Section # 4

Problem # 1

Find the area under the standard normal curve, where Z is between z = -0.43 and z = 0.78

Answer

P(-0.43 < z < 0.78) = P(z < 0.78) – P(z < -0.43) = 0.7823 – 0.3336

= 0.4487

Problem # 2

Suppose you must establish regulations concerning the maximum number of people who can occupy a lift. You know that the total weight of 8 people chosen at random follows a normal distribution with a mean of 550kg and a standard deviation of 150kg.

What’s the probability that the total weight of 8 people exceeds 600kg?

Answer

The mean is 550kg and we are interested in the area that is greater than 600kg.

z = ( x - μ ) / σ

Here x = 600kg, μ = 550kg, σ = 150kg

P(x > 600) = P ( z > ( 600 - 550 ) / 150)

= 1 – 0.6923 = 0.3707

Problem # 3

Molly earned a score of 940 on a national achievement test. The mean test score was 850 with a standard deviation of 100. What proportion of students had a higher score than Molly? (Assume that test scores are normally distributed.)

(A)0.10 (B)0.18 (C)0.50 (D)0.82 (E) 0.90

Answer

The correct answer is B. As part of the solution to this problem, we assume that test scores are normally distributed.

z = (X - μ) / σ

= (940 - 850) / 100

= 0.90

P(Z < 0.90) = 0.8159.

Therefore, the P(Z > 0.90) = 1 - P(Z < 0.90) = 1 - 0.8159 = 0.1841.

Thus, we estimate that 18.41 percent of the students tested had a higher score than Molly.

Problem # 4

A company pays its employees an average wage of $3.25 an hour with a standard deviation of 60 cents. If the wages are approximately normally distributed, determine the proportion of the workers getting wages between $2.75 and $3.69 an hour

Answer

P (2.75 < x < 3.69) = P ([2.75 – 3.25]/0.6 < z < [3.69 – 3.25]/0.6) = P (-0.83 < z < 0.73) = P (z < 0.73) – P (z < -0.83) = 0.7673 – 0.2033 = 0.5640

Problem # 5:

The average life of a certain type of motor is 10 years, with a standard deviation of 2 years. If the manufacturer is willing to replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the lives of the motors follow a normal distribution.

Answer

P (X < x) = P (Z < k) = 0.03

From the table:k = -1.88

Therefore: Z = (x - µ) / σ -1.88 = (x – 10)/2

X = -1.88*2+10 = 6.24 years