# MATHEMATICS

5.5 The number of arrivals per minute at a bank located in the central business district of a large city was recorded over a period of 200 minutes, with the following results:

Arrivals Frequency

0 14

1 31

2 47

3 41

4 29

5 21

6 10

7 5

8 2

a. Compute the expected number of arrivals per minute.

b. Compute the standard deviation.

5.10 If n=5 and (pie symbol) = .4, what is the probability that

a. X=4?

b. X (smaller than or equal to) 3?

c. X <2?

d. X >1?

5.15 When a customer places an order with Rudy’s OnLine Office Supplies, a computerized accounting information system (AIS) automatically checks to see if the customer has exceeded his or her credit limit. Past records indicate that the probability of customers exceeding their credit limit is 0.05. Suppose that, on a given day, 20 customers place orders. Assume that the number of customers that the AIS detects as having exceeded their credit limit is distributed as a binomial random variable.

a. What are the mean and standard deviation of the number

of customers exceeding their credit limits?

b. What is the probability that zero customers will exceed their limits?

c. What is the probability that one customer will exceed his or her limit?

d. What is the probability that two or more customers will exceed their limits?

5.19 Assume a Poisson distribution.

a. If l= 2.0, find P (X (greater than or equal to) 2)

b. If I= 8.0, find P (X (greater than or equal to) 3)

c. If I = 0.5, find P (X (smaller than or equal to) 1)

5.21 Assume that the number of network errors experienced

in a day on a local area network (LAN) is distributed

as a Poisson random variable. The mean number of network

errors experienced in a day is 2.4. What is the probability

that in any given day

a. zero network errors will occur?

b. exactly one network error will occur?

c. two or more network errors will occur?

d. fewer than three network errors will occur?

5.39 Errors in a billing process often lead to customer dissatisfaction and ultimately hurt bottom-line profits. An article in Quality Progress (L. Tatikonda, “A Less Costly Billing Process,” Quality Progress, January 2008, pp. 30–38) discussed a company where 40% of the bills prepared contained errors. If 10 bills are processed, what is the probability that

a. 0 bills will contain errors?

b. exactly 1 bill will contain an error?

c. 2 or more bills will contain errors?

6.1 Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1, as in Table E.2), what is the probability that

a. Z is less than 1.57?

b. Z is greater than 1.84?

c. Z is between 1.57 and 1.84?

d. Z is less than 1.57 or greater than 1.84?

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?