THE UNIVERSITY OF THE WEST INDIES

ST. AUGUSTINE

EXAMINATIONS OF May 2010

Code and Name of Course: MATH 2240 - Statistics for Engineers Paper:

Date and Time: Duration: 2 hours

INSTRUCTIONS TO CANDIDATES: This paper has 5 pages and 6 questions

ANSWER FOUR QUESTIONS INCLUDING AT LEAST ONE OF QUESTIONS FIVE OR SIX.

© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

1

Statistical Tables are provided.

Graph Paper is provided.

Non Programmable Calculators are allowed

THE UNIVERSITY OF THE WEST INDIES

ST. AUGUSTINE

EXAMINATIONS OF May 2010

Code and Name of Course: MATH 2240 - Statistics for Engineers Paper:

Date and Time: Duration: 2 hours

INSTRUCTIONS TO CANDIDATES: This paper has 5 pages and 6 questions

ANSWER FOUR QUESTIONS INCLUDING AT LEAST ONE OF QUESTIONS FIVE OR SIX.

© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

2

PLEASE TURN TO THE NEXT PAGE

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© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

2

1(i) Define the following statistical terms:

a. Outcome

b. Sample Space

c. Event

d. Random Variable

[4]

(ii) NIS monthly payments are based on the numbers of Mondays in the month. What is the probability that

a 31 day month will have five Mondays? (Assume that there is equal chance of any one of the seven

days being the first day of a month)

[6]

(ii) State Bayes Theorem

At a certain gas station, 40% of the customers use regular unleaded gas, 35% use extra unleaded gas and

25% use premium unleaded gas. Of those customers using regular gas, only 30% fill their tanks. Of

those customers using extra gas, 60% fill their tanks, whereas of those using premium, 50% fill their

tanks.

a. What is the probability that the next customer will request extra unleaded gas and fill the tank? State

the rule used.

b. What is the probability that the next customer fills the tank?

c. If the next customer fills the tank, what is the probability that regular gas is not requested?

[15]

2. (i) Let X denote the amount of space occupied by an article placed in a two cubic feet packing container. The

pdf of X is

2 0 2( )

0 otherwise

kx xf x

a. Find the value of k.

b. Obtain the cdf of X.

c. Compute ( ) and .xE X

[10]

(ii) “Time headway” in traffic flow is the elapsed time between the times that one car finishes passing a

fixed point and the instant that the next car begins to pass that point. Let X = the time headway for two

randomly chosen consecutive cars on a freeway during a period of heavy flow. The following pdf of X

is

Page

© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

3

.15 .5

.15 .5

0 otherwise

xe x

f x

What is the probability that the time headway is at most 6 seconds? [6]

(iii) In a large shipment of parts, 1% of the parts do not conform to specifications. The supplier inspects a

random sample of 30 parts, and the random variable X denotes the number of parts in the sample that do

not conform to specifications. The purchaser inspects another random sample of 20 parts, and the

random variable Y denotes the number of parts in this sample that do not conform to specifications.

What is the probability that X ≤ 1 and Y ≤ 1? [7]

(State any assumptions made.)

Is inspection an effective means of achieving quality? [2]

3(i) The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all

resistors having a resistance exceeding 10.634 ohms, and 5% having a resistance smaller than 9.7565

ohms. What are the mean value and standard deviation of the resistance distribution? [8]

(ii) The lifetime of a certain type of battery is normally distributed with mean value 12 hours and standard

deviation 1 hour. There are four batteries in a package. What lifetime value is such that the total

lifetime of all batteries in a package exceeds that value for only 5% of all packages? [7]

(iii) State three conditions necessary for a Poisson Process to occur. [3]

The number of tickets issued by a meter reader for parking-meter violations can be modeled by a

Poisson process with a rate parameter of five per hour.

a. What is the probability that exactly three tickets are given out during a particular hour? [5]

b. How many tickets do you expect to be given during a 45-min period? [2]

4(i) The accompanying data describe flexural strength (Mpa) for concrete beams of a certain type.

9.2 9.7 8.8 10.7 8.4 8.7 10.7

6.9 8.2 8.3 7.3 9.1 7.8 8.0

8.6 7.8 7.5 8.0 7.3 8.9 10.0

8.8 8.7 12.6 12.3 12.8 11.7

Page

© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

4

a. Calculate a point estimate of the mean value of strength for the conceptual population of all beams

manufactured in this fashion, and state which estimator you used. Hint: 246.8.ix

[2]

b. Calculate and interpret a point estimate of the population standard deviation . Which estimator did

you use? Hint: 2 2327.54.ix [4]

c. Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 11

Mpa. Hint: Think of an observation as a “success” if it exceeds 11. [2]

(ii) A random sample of n items manufactured by a certain company is selected. Let X = the number among

the n that are flawed and let p = P (item is flawed). Assume that only X is observed, rather than the

sequence of 's and 's.S F

a. Derive the maximum likelihood estimator of p . If n = 25 and x = 5, what is the estimate? [6]

b. Is the estimator of part (a) unbiased? [2]

c. If n = 25 and x = 5, what is the mle of the probability that none of the next five items examined is

flawed? [2]

(iii) A random sample of n = 8 E-glass fiber test specimens of a certain type yielded a sample mean

interfacial shear yield stress of 30.5 and a sample standard deviation of 3.0. Assuming that interfacial

shear yield stress is normally distributed, compute a 95% CI for true average stress. [7]

5(i) Light bulbs of a certain type are advertised as having an average lifetime of 800 hours. The price of

these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase

arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than

what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb determined,

and the appropriate hypotheses were tested using MINITAB, resulting in the accompanying output.

Variable n Mean St. Dev SE of Mean Z P-Value

Lifetime 50 738.44 38.20 5.40 -2.14 0.016

What conclusion would be appropriate for a significance level of .05? [5]

(ii) The desired percentage Si2O in a certain type of aluminous cement is 5.5. To test whether the true

average percentage is 5.5 for a particular production facility using a significance level of .01, 16

independently obtained samples are analyzed. Suppose that the percentage of Si2O in a sample is

Page

© The University of the West Indies Course Code: 200/200/……. ___________________________________________________________________________________________________________

DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY

INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner and/or

the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First

and Second Examiners. Completed forms should be handed to the Assistant Registrar (Examinations). The EXTERNAL

EXAMINER is requested to sign the question paper and return it with comments, it any, (on a separate sheet), to the Assistant

Registrar (Examinations).

……………………………………. ……………………………………..

First Examiner University Examiner

Date: 20…/…../….. Date: 20…../……/……

…………………………………… …………………………………….

Second Examiner External Examiner (where applicable)

Date: 20…/…../….. Date: 20…../……/……

5

normally distributed with .3 and that 5.25.x

a. Does this indicate conclusively that the true average percentage differs from 5.5? [6]

b. If the true average percentage is 5.6 and a level .01 based on n = 16 is used, what is the

probability of detecting this departure from 0 ?H [3]

(iii) Consider the accompanying data on breaking load (kg/25 mm width) for various fabrics in both an

unabraded condition and an abraded condition. Test at significance level .01 whether or not the true

mean breaking load is greater for unbraded condition than for abraded.

Fabric

1 2 3 4 5 6 7 8

U 25.6 48.8 49.8 43.2 38.7 55.0 36.4 51.5

A 26.5 52.5 46.5 36.5 34.5 20.0 28.5 46.0

[11]

6. The accompanying data on x = current density (mA/cm 2 ) and y = rate of deposition ( m/min) appeared

in a recent study.

x 20 40 60 80

y 0.24 1.20 1.71 2.22

a. Construct a scatter plot of the data. Does there appear to be a very strong relationship between the

current density and rate of deposition? [3]

b. Do you agree with the claim by the article’s author that “a linear relationship was obtained from the

tin-lead rate of deposition as a function of current density”? Explain your reasoning. [15]

c. What is the expected change in the rate of deposition for a change of 1 mA/cm 2 in the current

density? [2]

d. What is the expected deposition rate when the current density is 50 mA/cm 2 ? [5]

END OF QUESTION PAPER