Question

1. For a certain population the weights are normally distributed with a mean of 120 and a standard deviation of 15. Using the empirical rule, calculate the following probabilities: a. P(90 < weight< 150) b. P(weight > 135)

2. See exercise 1. Calculate the z-scores for the following weights, rounding to the nearest hundredth: a. 101 b. 142

3. See exercise 1. Calculate the weights for the following z-scores, rounding to the nearest integer: a. z = 2.22 b. z = 1.56 4. See exercise 1. What is the probability that a randomly selected subject will have a weight between 108 and 125?

Answer 1

1. Let X be the random variable denoting the weights.

Thus, X ~ N(120, 15) i.e. (X - 120)/15 ~ N(0,1)

a. P(90 < X < 150)

= P[(90 - 120)/15 < (X - 120)/15 < (150 - 120)/15]

= P[-2 < (X - 120)/15 < 2] = (2) - (-2) = 0.9772 - 0.0228

= 0.9544. (Ans).

b. P(X >135) = P[(X - 120)/15 > (135 - 120)/15]

= P[(X - 120)/15 > 1] = 1 - P[(X - 120)/15 < 1]

= 1 - (1) = 1 - 0.8413 = 0.1587. (Ans).

2. a. z-score of 101 is = (101 - 120)/15 = - 1.2667. (Ans).

b. z-score of 142 is = (142 - 120)/15 = 1.4667. (Ans).

3. a. Weight of z-score of 2.22 = 120 + (2.22 * 15) = 153.3.

(Ans).

b. Weight of z-score of 1.56 = 120 + (1.56 * 15) = 143.4. (Ans).

4. The required probability = P(108 < X < 125)

= P[(108 - 120)/15 < (X - 120)/15 < (125 - 120)/15]

= P[-0.8 < (X - 120)/15 < 0.3333] = (0.3333) - (-0.8)

= 0.6305 - 0.2119 = 0.4186. (Ans).