please show the work. 

Youmake your own copy to follow the discussions in class before the exam

Show all work (even for multiple choice questions, if appropriate) (Max 50 points)

1. (1 point) If P(A) = 0.3, P(B) = 0.8 and P(A or B) = 0.8, then P(A and B) is equal to ____________.

2. (1 point) If A and B are independent events with P(A) = 0.3 and P(B) =0.7, then P(A or B) = _____________.

3. (2 points) Explain the practical effect on your firm’s purchasing practice, if quality of the final product and quality of the raw material that go into the product are independent. Assume that you have three premier suppliers. Would you like to purchase from the lowest cost supplier or the medium or the highest cost supplier? Explain by first explaining the meaning of “independence” in this context.

4. (3 points) A quality engineer in a light bulb factory is planning a study to estimate the average life of a large shipment of light bulbs. The engineer wants to estimate the average life within ±15 hours with 95 percent confidence level. Assuming a process standard deviation of 25 hours, what is the sample size?

5. (3 points) Length of a confidence intervals for a population parameter, such as mean, is defined as the higher value – lower value. Example, if [3,7], the range is 7-3=4. In general, as other things remaining the same but,

a. As sample size (n) increases, Length of the interval will become smaller. True or False; Justify

b. As confidence level increases, length of the interval will become smaller. True or False; Justify

c. As mean shifts, (population s.d. remains the same) , no effect will be seen in the length of the interval. True or False; Justify

6. (6 points) A filling machine has a process mean (μ) of 300 OZ and s.d. (σ) of 2 OZ. It is controlled by a control chart with sample size 4 and k=3.

a. What is the UCL?

b. What is the LCL?

c. Specification for the bottles are 300±8 oz. and any bottle within that range is considered as properly filled bottle). What proportion of bottles will be under filled (below 292 oz) using the process described above?

d. Suppose that you are going to supply

100

0 bottles, assuming that Under-filled bottle attracts a $10 penalty, how much penalty would you be paying on an average, assuming that the customer can identify all under under-filled bottles.

e. Assuming: a. over-filing is not objected by the customer b. An OZ of the material you supply costs you $0.20: If you shift the mean up by 1.0 oz, what would be the overall cost of the shift, including savings in under-filled bottle penalty and the increase in the amount of material you will be using, for a batch of 1000. (Show normal distribution figures marking the mean, shift, etc.

7. (3 points) Historical production data indicate that the length of a steel rods is normally distributed with a mean of 100 cms and a standard deviation of 6 cm. Suppose that a sample of 36 steel rods are randomly selected from a very large lot. Show the distribution of the sample mean and the original distribution of the population of steel rods on the same graph. Use a proper scale to draw your normal distribution. (You will be drawing two (one for the population and one for the sample average) normal distributions to scale.

100

8. (7 points) Suppose that a population of ropes used to carry heavy weights in cranes should be such that the “good” quality ropes should have an average break strength of 1000 pounds. We would like the good quality ropes to be accepted 94% of the time using a sample size of n, and appropriate critical value. A “poor” quality rope will have a population mean break strength of 990 pounds. We would like this to be accepted only 2% of the time, using the same sample size. All populations have a standard deviation of 15 pounds. If the population has a higher breaking strength like 1100 pounds, it is considered great and obviously never rejected. Please answer the question below.

a. Draw a diagram and mark the two distributions, alpha, beta and CV.

b. What is the correct sample size that can achieve this? (leave the sample size as a fraction.)

c. What is the critical value? Verify your answer by calculating the critical value by both methods and check they are equal.

d. What is the relationship between alpha, beta and sample size? (specifically as sample size increases, what happens to alpha and beta) Discuss.

9. (7 points) You need 100 neon signs for a poker championship which willl last 75 hours. Vendor A offers signs which are expected to be work satisfactorily for 100 hours with a s.d. of 10 hours and the life of the sign follows normal distribution at a cost of $50 per sign. Vendor B offers signs that have an exponential life span with mean 300 at a cost of $20 per sign. Only 100 signs will be put up and burnt out signs will not be replaced. Assume that a burnt out sign costs you notionally, $100 per sign, which offer would you take? Justify with calculations of the total cost.

Vendor A: 

a. Cost of the signs:

b. Number of burnt out signs during the event.

c. Cost of the burnt out signs.

d. Total cost.

Vendor B: 

a. Cost of the signs:
b. Number of burnt out signs during the event.
c. Cost of the burnt out signs.
d. Total cost.

Which offer would you accept?

Why?

10. (6 points )What are the two different ways in which the term six sigma is used? (Explain)

11. (6 points)

Population A: 48% male and 52% female and both groups have similar interest in the preference for a policy of interest, that we are trying to measure.

Population B: 30% over 60 years of age; 50% between 30-60 and 20% between 18-30. Over 60 has a more uniform preference for a policy of interest, 30-60 has some variation and 18-30 has wide variations in their preference.

You are going to sample the populations to find a measure of interest in some characteristic. (example : preference for social policies).

a. In which population you think stratified sampling will have maximum benefit? Why?

b. Suppose you are sampling 100 people from population B, which sub-population will get disproportionately higher number of samples under stratified sampling? Why?

12. (5 points) Discuss the effect of changes in the sample size (increase and decrease) on average length (ARL) (will it go up or down or stay the same)

And

Discuss the effect of shifts in process mean on average length (ARL) (will it go up or down or stay the same).