Problem 1
A local photocopy shop has three black-and-white copy machines and two color copiers. Based on historical data the chances that each black-and-white copier will be down for repairs is 0.10. The color copiers are more of a problem and are down 20% of the time each.
(a)Based on this information, what is the probability that if a customer needs a color copy, both color machines will be down for repairs?
(b)If a customer wants both a color copy and a black-and-white copy, what is the probability that the necessary machines will be available? (assume that the color copier can also be used to make a black-and-white copy if needed)
(c)If the manager wants to have at least a 99% chance of being able to furnish a black-and-white copy upon demand, is the present configuration sufficient? (Assume that the color copier can also be used to make a black-and-white copy if needed). Back up your answer with appropriate probability computation.
Problem 2
For a number of years, the Seattle, Washington real estate market has been booming with prices skyrocketing. Recently, the Washington Real Estate Commission studied the sales patterns in Seattle for single-family homes. One chart presented in the commission’s report is reproduced here below. It shows the number of homes sold by price range and number of days the home was on the market:
Days on the MarketPrice Range ($000) / 1-7 / 8-30 / Over 30
Under $200 / 125 / 15 / 30
$200-$500 / 200 / 150 / 100
$501-$1,000 / 400 / 525 / 175
Over $1,000 / 125 / 140 / 35
(a)Using the relative frequency approach to probability assessment, what is the probability that a house will be on the market more than 7 days?
(b)Is the event 1-7 days on the market independent of the prices $200-$500?
(c)Suppose a home has just sold in Seattle and was on the market less than 8 days,
what is the most likely price range for that home?
Problem 3
The employee benefit research institute reports that in 2005 69% of workers reported that they and/or their spouses had saved some money for retirement.
(a)If a random sample of 30 workers is taken, what is the probability that fewer
than 17 workers and/or their spouses have saved some money for retirement?
(b)If a random sample of 50 workers is taken, what is the probability that more than 40 workers and/or their spouses have saved some money for retirement?
Problem 4
An Internet bank can process a maximum of 25 electronic transfers every minute during the busiest periods of the day. If it receives more transfer requests than this, then the bank’s computer system will become so overburdened that it will slow to the point that no electronic transfers can be handled. If, during the busiest periods of the day, requested for electronic transfers arrive at the rate of 170 per 10-minutes period on average, what is the probability that the system will be overwhelmed by requests? Assume that the process can be described using a Poisson distribution.
Problem 5
The Bryce Brothers Lumber Company is considering buying a machine that planes lumber to the correct thickness. The machine is advertised to produce “6-inch lumber” having a thickness that is normally distributed, with a mean of 6 inches and standard deviation of 0.1 inch.
(a)If building standards in the industry require a 99% chance of a board being between 5.85 and 6.15 inches, should Bryce Brothers purchase this machine? Why or why not?
(b)To what level would the company that manufactures the machine have to reduce the standard deviation for the machine to conform to industry standards?
Problem 6
A manager of 20-room motel estimates that 9% of all confirmed reservations are “no-shows”. Consequently, the hotel accepts confirmed reservations for as many as 25 rooms. If more confirmed reservations arrive than there are rooms, the overbooked guests are sent to another motel and given a complimentary dinner. If the motel currently has 25 confirmed reservations, find:
(a)The probability that no customers will be sent to another motel.
(b)The probability that exactly 2 guests will be sent to another motel.
(c)The probability that 3 or more guests will be sent to another motel.
Problem 7
The director of a state agency believes that the average starting salary for clerical employees in the state is greater than $30,000. per year. To test her hypothesis, she has collected a simple random sample of 100 starting clerical salaries from across the state and found that the sample mean is $30,250.
(a)State the appropriate null and alternative hypotheses
(b)Assuming the population standard deviation is known to be $2,500 and the
significance level for the test is to be 0.05, what is the critical value (stated in dollars)?
(c)Referring to your answer in part b, what conclusion should be reached with respect to the null hypothesis?
(d)Calculate the p-value.
Problem 8
The federal communications commission released a report in 2008 refuting an earlier report released in 2006. The report 2008 indicates that cable subscribers would save as much as 13% on their cable TV bills. The average monthly cable prices were estimated to be $41.04. Typically, such reports announce a margin of error of say, $1.25 and a confidence level of 95%. Suppose that the standard deviation of the monthly cost of cable TV bills was $10.00.
(a)Determine the sample size of the report released in 2008
(b)Calculate the sample size required to decrease the margin of error by a dollar
(c)A typical sample size used in national surveys is 1500 to 2000. Determine a range for the margin of error corresponding to this range of sample sizes.