2. CentralLimit Theorem (4)

1% of apopulation of manufactured parts are defective as a routine outcome of theprocess.  The quality control technicianperiodically samples 100 parts to determine if the process is in control.

a.  What is the sample proportion, P_bar,  of defective parts for n=100?

b.  What is the standard deviation of thesampling distribution, s_p-bar for n=100?

A populationdistribution has a mean of 99 with a standard deviation of 4,  Consider the sampling distribution forsamples of size 64,

c.  What is the mean of the samplingdistribution?

d.  What is the standard deviation of thissampling distribution?

3.  Estimation (4)

A random sample of 900 taxpayers in Maine was taken todetermine the average state income tax paid by these citizens.  A 95.5% confidence interval estimate wasconstructed and found to be:

P(200 < m  <  1400) = 95.5%

a. Withthat knowledge alone, what is the point estimate of the population average taxpaid?

b. Whatis the Z-score is associated with a 95.5% level of confidence?

c. Whatis the standard deviation of the sampling distribution for samples of size, n =900?

 d. What is the population standard deviationfor all taxpayers in Maine?

4.  Normal distribution (4) 

In a particular area of theNortheast, an estimated 75% of the homes use heating oil as the principalheating fuel during the winter.  A randomtelephone survey of 150 homes is taken in an

attempt to determine whether thefigure is correct.  Suppose 120 of the150 homes surveyed use heating oil as the principal heating fuel.  What is the probability of getting a sampleproportion this size or larger if the population estimate is true?

5.  Hypothesis Testing (8) 

Marta’s Bakery produces a boxedcake mix.  The boxes are filled by anautomatic device.  The filling process isnormally distributed and the average content of the boxes is set to be 20 oz. witha process standard deviation, ∂,  of 1oz.  In order to check on the adjustmentof the machine filling process we decide to take samples of 9 boxes and

                                    If19.5  <   X_bar  < 20.5  do not adjust the machine

                                    If  X_bar  < 19.5   or   X_bar  > 20.5   adjust the machine

a.     Statethe null and alternate hypotheses.

b.     Whatis the standard error of the mean, ∂_x-bar?

c.     Whatis the probability of a Type I error?

d.     Ifthe average fill of each box has actually shifted to 21 oz., what is theprobability we

            willfail to have the machine adjusted?  Thatis, what is the Type II error probability?

6. Hypothesis Testing (5)

Throughout the 1990’s theaverage household expenditure on scratch tickets in Maine has been $5.  In 2010, a sample of 7 households was selectedat random and the following expenditures for a given week were reported:

                                          $3,$2, $4, $3, $5, $4, $7 

a. What is the sample mean expenditure for scratch tickets?

b. What is the sample standard deviation for the sampleresults?

c. What are the null and alternate hypotheses if we wishedto test the hypothesis that the average  

     expenditure onscratch tickets has decreased in the 21^st century?

d.  What is thecritical score which will determine your conclusion?

e.  What is thecomputed score found from the sample data and what is your conclusion based on

     these results?

7. Concepts (5)

a.  With respect to the central limit theorem,what is the mathematical relationship between the mean and standard deviation of the underlying parent populationand the mean and standard deviation of a sampling distribution of the mean fora given size sample?

QSO510, p3

b.  Regarding data, why do we most often utilizesample information and summary data instead of analyzing census data?

c. Briefly explain what is thetype I, alpha error, and the type II, beta error.

d.  In a given situation the power of the test is95 percent.  What does that tell us?

e.  In a world of seemingly random happenstance,how is it that we are able to place events of all sorts in the context of anormally distributed population and make reasonably accurate statementsregarding the populations from which they were derived?

8. Linear Programming (7 points)

A small winery manufactures 2types of wine, Burbo's Better (X) and Burbo's Best (Y).  Burbo's Better results in profit of $4 perquart, whereas Burbo's Best has profit of $5 per quart.  Two production workers mix the 2 wines.  It takes a production worker 2 hours to mix aquart of the Better and 3 hours to mix a quart of the Best.  Each worker puts in a 9 hour day.  The quantity of alcohol than can be used tofortify the wine is limited to 24 ounces daily. Six ounces of alcohol are added to each quart of Burbo's Better and 3ounces are added to Burbo's Best.

a. (1 point) State the objectivefunction.

b. (2 points)  State the constraint functions.

c. (2 points)  Graph the constraints and identify thefeasible area.

d. (2 points)  Evaluate the relevant production points andidentify the production schedule that will maximize the company’s profit.

9. Decision Making  (8 pts.)

Ernie Schlock is one of Boston'sforemost  new car dealers.  Ernie has always been proud of his ability toaccurately assess the market and thereby make the right decisions when it cameto promoting the newest models.  However,this year has been exceptionally difficult because of rising fuel costs.  Ernie is especially unsure of the market forhis new line-up of hybrid automobiles. If fuel prices skyrocket and he promotes the hybrid aggressively, hisprofits on the sales of hybrids will be $800,000.  On the other hand, asoft promotion of these cars will lead to profits of only 200,000.  A middle approach to promotion will yieldprofits of $500,000.   Now if the priceof fuel drops below $3/gallon, Ernie can expect to suffer some significantlosses because the auto buying public prefers the gas guzzling dinosaurs inspite of their inevitable future demise. In this case, Ernie estimates he will lose about $500,000 with anaggressive promotion of hybrids and incur about a $100,000 loss with a softpromotion.  A middle ground promotionwill also result in a $200,000 loss for Mr. Schlock.  There is also the possibility that gas priceswill hover around $4/gallon for the remainder of this year in which case anyincremental sales from advertising will be offset by the cost of promotion andhe feels that no matter how aggressively he promotes his hybrids, profits willbe the same, at about $300,000

                                                            FUELCOSTS

                                    Falling                         Stable                          Rising

P      Aggressive          -500                             300                              800

R

O      Middle               -200                             300                              500

M

O       Soft                   -100                             300                              200

a. Evaluate this situation forErnie utilizing the maximax, maximin, LaPlace, and the minimax criterion. Whatis your overall recommendation to Mr. Schlock?

b.  Suppose Ms. Esta Mator has studied the global oil markets and informs you thatthere is a 20% chance of stable fuel costs, while the likelihood of eitherfalling or rising costs is about the same. Evaluate the decision for Mr. Schlock using expected value analysis.

c.  Esta's partner, Sue Sayer, claims to haveperfect knowledge of the world oil futures market and is willing to share herknowledge with you for a price.  What isthe maximum you would be willing to pay her at the point of being indifferent?