IOE 265



IOE 265 Midterm II Winter ‘03

Name (printed): ________________________________

Student ID #: ____________________________

Lab Section # (circle): Sec 2 Sec 3 Sec 4

TU 12:30-1:30 TU 1:30-2:30 TU 2:30-3:30

Sec 5 Sec 6 Sec 7 Sec 8

TU 3:30-4:30 TU 4:30-5:30 W 9:30-10:30 W 10:30-11:30

“I have neither given nor received aid, nor have I concealed any violation of the Honor Code.”

Signature: _____________________________

You are allowed to use a single crib sheet for this exam. The crib sheet should be 8.5” x 11” and you can have information on one side of the sheet.

There are 10 written pages to this exam, including the cover page. Please check now to make sure that you have the appropriate number of pages. Put your name at the top of each page. Pace yourself on the exam. Good luck!

|Part |Points Possible |Your Points |

|1 – Multiple Choice |30 | |

|2 – Problem 1 |15 | |

| Problem 2 |15 | |

| Problem 3 |15 | |

| Problem 4 |15 | |

| Problem 5 |15 | |

|TOTAL |105 | |

Total Points

Part I. Multiple Choice. (30 points).

Read each question carefully. Circle the answer – a,b,c, or d – which you feel best answers the question. Each question is worth 5 points. No partial credit will be given for questions 2 through 6. Partial credit will be given for question 1 (3 points for correct choice and 2 points for additional statement).

1. A random sample of 20 observations was selected and the following probability plots were constructed in Minitab. What is the most plausible distribution family from which the given sample was selected? Please include a statement to support your choice.

a. Exponential

b. Normal

c. Lognormal

d. Weibull

[pic]

Points fall closest to a straight line on the lognormal probability plot

2. Which of the following is the best estimate for the variance of the sample mean statistic X-bar?

a. [pic]

b. [pic]

c. [pic] = S2/n is the correct choice

d. [pic]

3. Consider recording the number of defects per hour found by two quality inspectors, 1 and 2, respectively. The two populations of defect rates analyzed by the two inspectors each follow a Poisson distribution with parameters, λ1 = 5 and λ2 = 4, respectively. Two independent samples of sizes n1=35 and n2=45, respectively, are selected from the given populations. What is the probability that the estimator for the difference between the two sample means,[pic] is less than or equal to two defects?

Use the Central Limit Theorem, both n > 30

P([pic]( 2 ) = [pic] = [pic] = [pic]= 0.9812

a. 0.1774

b. 0.9812

c. 0.5997

d. 0.6293

4. A sample of n observations is selected from a population with known pdf f(x;θ)=θe-θx. If the method of moments is used to find an estimate for θ, which of the following equation(s) must be solved?

a. [pic]

b. [pic] and [pic]

c. [pic]

d. Both a and c

5. Assume the weights of individuals are independent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people try to squeeze into an elevator. What is the design limit of this elevator such that only 0.02% of the time the load (total weight) exceeds the design limit?

a. 4230.9 lbs

b. 4523.5 lbs

c. 3784.2 lbs

d. 4500 lbs

Design Limit = DL, T0 = load (total weight)

P(T0 > DL) = 0.0002, where T0 ~ N(nμ,σ*sqrt(n)) = N(4,000;150)

So DL is 99.98th percentile of T0 = 4,000 + 150 * 99.98th percentile of Z = 4,000 + 150 * 3.49 = 4,523.5 lbs

6. Upon completion of a marketing campaign, an advertising agency claimed that 20% of customers in a state had become aware of the newly advertised product. The product’s distributor then surveyed a random sample of 1,000 of customers in the state. Assuming that the ad agency’s claim is true, what is the probability that between 185 and 220 (inclusive) of the selected customers in the state had become aware of the newly advertised product?

a. 0.8381

b. 0.6724

c. 0.0567

d. 0.1769

There is an underlying binomial distribution with n = 1000 and p = 0.20. However, since np ≥ 10 and

nq ≥ 10, we can use the normal approximation

[pic]

Thus,

[pic]

[pic]

[pic]

[pic]

Part II. Problems (15 points each)

Solve the following problems. Show all your work and clearly mark your answer.

1. The continuous random variables X and Y have the joint pdf:

fxy(x,y) = [pic]

Also, it can be shown that fy(y) = 4y(1-y2) for 0≤y≤2 and E(Y) = 8/15.

Note: if you need to perform a double sum or integral, it may be more convenient to sum or integrate over y before summing or integrating over x.

a. Find the value of c.

[pic]

[pic]

c = 3/16 = 0.1875.

b. Find fx(x), the marginal distribution of X.

fx(x) = [pic]

c. Find E(X).

E(X) = [pic]= 63/40 = 1.575

d. Are X and Y independent? Why or why not?

[pic], so X and Y are not independent.

e. Find the covariance of X and Y.

E(XY) = [pic]

[pic]

[pic]

2. While taking an IOE 265 exam, previous students have reported an interesting phenomenon. Students have noted that upon turning to each new page of the exam there is the possibility of having a panic attack. Given a 7-page exam, the data below was gathered on the number of panic attacks experienced by thirty students. (Assume that the probability of having a panic attack is constant and is identical for each student.)

x1 =1 x2 =1 x3 =1 x4 =4 x5 =3 x6 =2 x7 =2 x8 =0 x9 =1 x10=1

x11=2 x12=2 x13=0 x14=2 x15=1 x16=2 x17=2 x18=1 x19=3 x20=2

x21=1 x22=0 x23=1 x24=3 x25=2 x26=0 x27=1 x28=1 x29=2 x30=2

Hint: [pic] [pic]

a. What is the maximum likelihood estimate (MLE) of the probability of having a panic attack on any given page? Show all steps involved in the MLE process.

The pmf for a binomial distribution is given by, [pic]

Thus, the likelihood function is given as follows:

[pic]

[pic]

The maximum of this function is determined by taking the first derivative of the natural log of the likelihood function and setting it equal to zero,

[pic]Thus,

[pic]

Note: These thirty data were randomly generated from a binomial distribution with n=7 and p=0.25 using Minitab.

b. What is the maximum likelihood estimate (MLE) of the probability of a student having his/her second panic attack on page 5 of the 7-page exam? Name the principle you used to find this MLE

Due to the invariance principle, we can simply use the negative binomial distribution with the mle estimate of p determined above.

[pic]

Thus,

[pic]

3. Suppose that the number of accidents occurring in an industrial plant is described by a Poisson distribution with an average of 6 accidents per year. Let X denote the time (in months) between successive accidents.

a. Find the probability density function of X.

Poisson rate 6 accidents per year = 0.5 accidents per month … X ~ EXP(0.5)

Density f(x) = ½ exp(-x/2) for x≥0

b. Using the pdf of X, find the expected time (in months) between successive accidents. Is this result consistent with the initial statement given in this problem? Why or why not?

E(X) = 1/λ = 1/0.5 = 2, consistent … average of 6 accidents for 12 months … expect 2 months btw successive accidents

c. Find the probability that the time between successive accidents is more than 4 months.

P(T>4) = 1-FX(4)=1-(1-exp(-4/2))=exp(-2)=0.1353

d. Find the probability that the time between successive accidents is between 1 and 3 months.

P(1 ................
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