# MATHEMATICS

6.5 Given a normal distribution with m=100 and s= 10, what is the probability that

a. X > 75?

b. X < 70?

c. X < 80 or X > 110?

d. Between what two X values (symmetrically distributed around the mean) are 80% of the values?

6.9 Consumers spend an average of $21 per week in cash without being aware of where it goes (data extracted from “Snapshots: A Hole in Our Pockets,” USA Today, January 18, 2010, p. 1A). Assume that the amount of cash spent without being aware of where it goes is normally distributed and that the standard deviation is $5.

a. What is the probability that a randomly selected person will spend more than $25?

b. What is the probability that a randomly selected person will spend between $10 and $20?

c. Between what two values will the middle 95% of the amounts of cash spent fall?

6.11 A statistical analysis of 1,000 long-distance telephone calls made from the headquarters of the Bricks and Clicks Computer Corporation indicates that the length of these calls is normally distributed, with m = 240 seconds and s = 40 seconds.

a. What is the probability that a call lasted less than 180 seconds?

b. What is the probability that a call lasted between 180 and 300 seconds?

c. What is the probability that a call lasted between 110 and 180 seconds?

d. 1% of all calls will last less than how many seconds?

6.13 Many manufacturing problems involve the matching of machine parts, such as shafts that fit into a valve hole.

A particular design requires a shaft with a diameter of 22.000 mm, but shafts with diameters between 21.990 mm and 22.010 mm are acceptable. Suppose that the manufacturing process yields shafts with diameters normally distributed, with a mean of 22.002 mm and a standard deviation of 0.005 mm. For this process, what is a. the proportion of shafts with a diameter between 21.99 mm and 22.00 mm?

b. the probability that a shaft is acceptable?

c. the diameter that will be exceeded by only 2% of the shafts?

d. What would be your answers in (a) through (c) if the standard deviation of the shaft diameters were 0.004 mm?

6.29 An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inch. The lower and upper specification limits under which the ball bearings can operate are 0.74 inch and 0.76 inch, respectively. Past experience has indicated that the actual diameter of the ball bearings is approximately normally distributed, with a mean of 0.753 inch and a standard deviation of 0.004 inch. What is the probability that a ball bearing is

a. between the target and the actual mean?

b. between the lower specification limit and the target?

c. above the upper specification limit?

d. below the lower specification limit?

e. Of all the ball bearings, 93% of the diameters are greater than what value?