# Statistics

**Question description**

**2.** **Central
Limit Theorem **(**4) **

1% of a population of manufactured parts are defective as a routine outcome of the process. The quality control technician periodically 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 the
sampling distribution, s_{p-bar} for n=100?

A population distribution has a mean of 99 with a standard deviation of 4, Consider the sampling distribution for samples of size 64,

c. What is the mean of the sampling distribution?

d. What is the standard deviation of this sampling distribution?

**3. Estimation (4)**

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

**P(200 <
****m < 1400) = 95.5%**

** **

a. With that knowledge alone, what is the point estimate of the population average tax paid?

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

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

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

** **

** **

** **

** **

**4. Normal distribution (4)**

In a particular area of the Northeast, an estimated 75% of the homes use heating oil as the principal heating fuel during the winter. A random telephone survey of 150 homes is taken in an

attempt to determine whether the figure is correct. Suppose 120 of the 150 homes surveyed use heating oil as the principal heating fuel. What is the probability of getting a sample proportion this size or larger if the population estimate is true?

** **

**5.** ** Hypothesis Testing (8)**

Marta’s Bakery produces a boxed cake mix. The boxes are filled by an automatic device. The filling process is normally distributed and the average content of the boxes is set to be 20 oz. with a process standard deviation, ∂, of 1 oz. In order to check on the adjustment of the machine filling process we decide to take samples of 9 boxes and

If
19.5 < X_{bar} <
20.5 do not adjust the machine

If X_{bar} <
19.5 or X_{bar} > 20.5
adjust the machine

a. State the null and alternate hypotheses.

b. What
is the standard error of the mean, ∂_{x-bar}?

c. What is the probability of a Type I error?

d. If the average fill of each box has actually shifted to 21 oz., what is the probability we

will fail to have the machine adjusted? That is, what is the Type II error probability?

**6. Hypothesis Testing (5)**

Throughout the 1990’s the average household expenditure on scratch tickets in Maine has been $5. In 2010, a sample of 7 households was selected at 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 sample results?

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

expenditure on
scratch tickets has decreased in the 21^{st} century?

d. What is the critical score which will determine your conclusion?

e. What is the computed 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 population and the mean and standard deviation of a sampling distribution of the mean for a given size sample?

QSO510, p3

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

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

d. In a given situation the power of the test is 95 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 a normally distributed population and make reasonably accurate statements regarding the populations from which they were derived?

8. Linear Programming (7 points)

A small winery manufactures 2 types of wine, Burbo's Better (X) and Burbo's Best (Y). Burbo's Better results in profit of $4 per quart, 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 a quart 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 to fortify the wine is limited to 24 ounces daily. Six ounces of alcohol are added to each quart of Burbo's Better and 3 ounces are added to Burbo's Best.

a. (1 point) State the objective function.

b. (2 points) State the constraint functions.

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

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

9. Decision Making (8 pts.)

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

FUEL COSTS

Falling Stable Rising

P Aggressive -500 300 800

R

O Middle -200 300 500

M

O Soft -100 300 200

a. Evaluate this situation for Ernie utilizing the maximax, maximin, LaPlace, and the minimax criterion. What is your overall recommendation to Mr. Schlock?

b. Suppose Ms. Esta Mator has studied the global oil markets and informs you that there is a 20% chance of stable fuel costs, while the likelihood of either falling 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 have perfect knowledge of the world oil futures market and is willing to share her knowledge with you for a price. What is the maximum you would be willing to pay her at the point of being indifferent?

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