Question

David sells ice cream at a local playground. His average daily income is $200 with a standard deviation of $30. Assume David's earnings are normally distributed.

a) David made $180 today. Based on his past earnings, what is the z-score for $180? (Round to no less than two decimal places.

b) What is the probability that David will make $180 or more tomorrow? (Give the proportion correct to four decimal places.)

Answer 1

X : David's earnings

X normally distributed with mean : $200 and standard deviation : $30

a)

David made $180 today. Based on his past earnings, what is the z-score for $180

Z-score for a given X = (X - mean)/Standard deviation = (X-200)/30

z-score for $180 = (180-mean)/Standard deviation = (180-200)/30=-20/30 = -0.67

z-score for $180 = -0.67

b)

probability that David will make $180 or more tomorrow = P(X > 180)

P(X > 180) = 1 - P(X 180)

Z; Standard normal distribution.

P(X 180) = P(Z   z-score of 180)

From a) z-score for $180 = -0.67

P(X 180) = P(Z   z-score of 180) = P(Z-0.67)

From standard normal tables,

P(Z-0.67) = 0.2514

P(X > 180) = 1 - P(X 180) = 1-0.2514 = 0.7486

probability that David will make $180 or more tomorrow = P(X > 180) =0.7486

Probability that David will make $180 or more tomorrow = 0.7486