# please see attached 10 questions

## Project Description:

1. following is the distribution for annual spend for a business unit in 2010

annualspend
(\$thousands) percentageof
contracts
0to10 18.0%
10to15 21.0%
15to25 7.0%
25to45 21.0%
45to65 13.0%
65to85 8.0%
85to120 4.0%
120to150 3.0%
150to250 3.0%
250to500 1.0%
500to1000 0.5%
1000to10,000 0.5%

a. what is the circa median spend for 2010?
b. what is the circa mean spend for 2010?
c. if a contract is picked at random from this group, what are the chances that the amount of spend on that contract is \$500,000 or less?
d. given we know the spend on a particular agreement was over \$120k, what are the chances the spend on that contract was over a million dollars?
e. if the spend on a contract is between \$25k and \$85k, what is the probability the spend is less than \$45k (that is, what proportion of spend in this range is on contracts with less than \$45k in 2010)?
f. what is the approximate shape of the distribution of spend for this business unit?
2. aexample of contracts is taken from the business unit in problem 1 and the age of each contract (how long that supplier has been under contract) and the exact amount of spend on the contract is recorded for each contract in the sample. the correlation between age and amount of spend is calculated.
a. if they find out after calculating the correlation that ages were recorded in months, not years, and the age values are all divided by 12 to change them to years, is it possible to tell what this will do to the correlation coefficient? if so, what?
b. if they add \$5000 to the spend amount for each contract in the sample to compensate for inspection fees paid for each contract but not recorded, is it possible to tell what that will do to the correlation coefficient? if so, what?

3. the number of bidders responding to an rfp for large-scale deep-water piping over the past 10 years has been either 3, 4, 5, or 6 bidders. the proportions of rfps with the numbers of bidders shown are 0.15, 0.25, 0.35, and 0.25, respectively. for example, 15% of the rfps attracted 3 bidders.
a. if x represents the number of bidders for an rfp, what is the probability distribution for x?
b. what is the probability that an rfp attracts 4 or 5 bidders?
c. what is the probability that an rfp selected at random attracted more than the average number of bidders?
d. three rfps are issued simultaneously and it is believed that the number of bidders for each is independent of the number for the other two. what is the probability that the total number of bidders for all three rfps (the sum of bids for rfp 1 and 2 and 3) is greater than 12?

4. a new moveable (“home”) diagnostic test for hiv was recently released and has been advertised this past week. the test, oraquick advance, states in the description of the test’s accuracy that it has “greater than 99%” compliance with a definitive test, so the new home test spots 99.5% of cases that are hiv positive (that is, if a person is hiv positive, the test spots this 99.5% of the time). it has, according to another publication, a false positive rate of about 1/12, or (for the purposes here) 8%. it is estimated that 0.3% of the u.s. population is hiv positive.
a. if a person is selected from the population at random and the oraquick test gives a positive test result for that person, what is the probability that person is actually hiv positive?
b. what proportion of the people who take the oraquick test will get positive test results (either rightly or wrongly)?
5. during the first 6 months of an sap installation, the number of “holds” on payments to suppliers skyrocketed. the length of time that these payments were on hold until the discrepancy was resolved was normally distributed with a mean of 43 days and a standard deviation of 8 days.
a. what proportion of holds were unresolved after 60 days?
b. what proportion of holds were on hold for between 30 and 50 days?
c. if you want to tell a supplier that there is 90% confidence that their hold will not take more than x number of days to be resolved, what is the value of x in days that makes this statement true?

6. a big factory wants to inspect the relationship between annual salaries (y) and the number of years employees have worked (x) at the factory. data was collected for a sample of 27 employees. the excel output is shown below:

summary measures

multipler 0.7952
r-square 0.6324
sterrofestimate 6595.19

anova table

source df ss ms f p-value
explained 1 18708543281870854328 43.012 0.0000
unexplained 25 1087414471 43496579

regression coefficients

coefficient stderr t-value p-value 95%
interval
constant
years employed 28326.88 2096.929 13.5087 0.0000
1067.03 162.699 6.5583 0.0000
843to
1281

a. what is the equation describing the relationship between salary and years worked?
b. is the relationship statistically significant? that is, does the data indicate there is actually a linear relationship between salary and seniority? why or why not?
c. give a 95% confidence interval for the average annual raise employees get at this factory.
d. what percentage of the variation in salaries at the factory is explained by variation in the number of years worked by employees?

7. the manager of a drilling well is interested in predicting the amount of hours of overtime per week that a person will voluntarily put in to get projects done. the explanatory variables are age (in years), education (highest level obtained, in years) and family size (number of people in household). the multiple regression output is shown below:

anova
table

source dfss ms f p-value
explained
unexplained 3 13.9682 4.6561 14.8564 0.0000
18 5.6413 0.3134

regression coefficients

coefficient stderr t-value p-value
constant
age education familysize 1.683 1.1696 1.4389 0.1674
-0.0498 0.0199 -2.5018 0.0222
0.2135 0.0503 4.2426 0.0005
0.0405 0.0784 0.5168 0.6116

a. does the model as a whole (with all 3 independent variables) explain a statistically significant proportion of the variation in the dependent variable? provide explanation for your answer.
b. give an estimate of the hours of overtime per week by a single 30-year-old college graduate (16 years of education).
c. are all three independent variables significantly related to the dependent variable at the .05 level? if not, which are not?
8. you are accountable for fuel purchasing for flights to an off-shore rig over the next year. the annual consumption of fuel to support these flights is 100,000 gallons. you can wait until october to buy the fuel in bulk at market prices; the price forecast is that there is a 60% chance the price will be \$5 per gallon and a 40% chance it will be \$7 per gallon. you have another alternative: you can pay now to have the option (the right but not the obligation) next october to pay \$5.50 per gallon; whatever you pay now to lock in this option you don’t get back. what is the most, if anything, you should be willing to pay for this option?
9. forecasting the price of spares containing high % of precious metals is complicated because both the costs and the volatility of costs are high. some modelling has been done to try to make this forecasting more accurate, but it is expensive and takes more time, which reduces the overall savings of using the forecasting models. below is the set of options for forecasting, the estimated volatility over the next purchasing phase, and estimated savings in contracts under each forecasting option (in \$ thousands).

low medium high very high
nomodeling 20 10 0 -30
simple modeling 10 10 10 -10
sophisticated
modeling -5 5 20 30
probability .2 .3 .4 .1
note: amountsinthe table areamountssavedin\$thousands,sonegative numbersorlosses.

a. which modeling strategy yields the greatest expected savings?
b. if it is critical that the business unit save \$20k or more, which strategy is best?
c. if a clairvoyant consultant is hired who can forecast the volatility future perfectly, what is the maximum you would be willing to pay for this information?

10. 1 out of every 20 shipments from a large warehouse (“fulfillment center” as they are now called in the retail trade) has some portion of the address or bar code inaccurate. these inaccuracies can result in delays in delivery of these items. if a shipment has an address inaccuracy, there is a 40% chance the driver can immediately detect the problem, make the necessary correction and there is no delay. there is a 40% chance there will be a delay of a day and there is a 20% chance there will be a delay of more than a day.
a. if 5,000 shipments go out of this distribution center a day, what is the expected number of shipments that have an inaccuracy in their labeling?
b. what is the probability that more than 400 shipments in a day have inaccuracies?
c. what is the expected number of shipments each day that are delayed more than a day in their delivery?
d. what is the expected amount of delay in shipments from this center (counting no delay as 0) in days?
e. what is the probability that any one shipment filled at this center will be delivered on time?
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