Stats Worksheet.docx

MODULE: ENGINEERING STATISTICS (STA 2023) WORKSHEET # 1

1.

The symmetric difference between two events A and B is the set of all sample points that are in exactly one of the sets and is often denoted AB .Note that AB  ( A  B ')  ( A ' B) Prove that P( AB)  P( A)  P( B)  2 P( A  B)

2.

Hydraulic assemblies for landing gear produced by an aircraft rework facility are inspected for defects. History shows that 8% have defects in the shafts alone, 6% have defects in the bushing alone, and 2% have defects in both the shafts and bushings. If one such assembly is randomly chosen, find the probability that it has the following characteristics: (i) (iii)

3.

(ii) (iv)

a shaft or bushing defect no defects in shafts or bushings

A bridge can be damaged by failure in the foundation (F) or in the superstructure (S). The corresponding failure probabilities for a particular bridge are estimated to be 0.05 and 0.01, respectively. Also, if there is foundation failure, then the probability that the superstructure will also suffer some damage is 0.50. (i) (ii)

4.

a bushing defect only one of the two types of defects

What is the probability of damage to the bridge? If F and S are statistically independent, what is the probability of damage to the bridge?

Relays in a section of an electrical circuit operate independently, and each one closes properly with probability 0.9 when a switch is thrown. The following two designs, each involving four relays, are presented for a section of a new circuit. A

C

a

b B

Fig. 1 (a)

D

A

C

a

b B

Fig.1 (b)

D

Which has the higher probability of current flowing from a to b when the switch is thrown?

1

5.

If A and B are two independent events, i.e. P( A  B)  P( A) P( B) then prove that P( A '  B ' )  P( A ' ) P( B ' )

6.

Show that

7.

If Aj ,

8.

A community is concerned about its power supply for the coming winter. There are three major sources of power supply, namely electricity, gas and oil. Let E, G, and O denote the events of shortage of each of these power sources, respectively. Their probabilities are estimated to be 0.15, 0.1 and 0.2 respectively. Furthermore, assume that if there is a shortage in the oil supply, the probability of an electrical power shortage will be doubled, that is, twice the probability of P(E) . The shortage of gas may be assumed to be independent of shortages of oil and electricity.

P[( A  B) / C ]  P( A / C )  P( B / C )  P( A  B / C )

n  n  j  1, 2,3,.........n are independent events, show that P  Aj   1   P( Aj ' ) j 1  j 1 

(i)

what is the probability that there will be a shortage of all three major sources of power supply?

(ii)

what is the probability that a shortage will occur in at least one of the following sources: gas, electricity?

(iii)

if there is a shortage of electricity, what is the probability that gas and oil also will be scare?

9.

The probability that a single aircraft engine will fail during flight is q. A multi-engine plane makes successful flight if at least half its engines run. Assuming that the engines operate independently, find the values of q for which a two-engine plane is to be preferred to a four-engine plane.

10.

A garage mechanic keeps a box of good springs to use as replacements on customers cars. The box contains 5 springs. A colleague, thinking that the springs are for scrap, tosses three faulty springs in to the box. The mechanic picks two springs out of the box while servicing the car, one at a time, without replacement. Find the probability that:

11

(i)

the first spring drawn is faulty

(ii)

the second spring drawn is faulty.

An assembler of electric fans uses motors from two sources. Company A supplies 90% of the motors and company B supplies the other 10%. Suppose it is known that 5% of the motors supplied by company A are defective and 3% of the motors supplied by company

2

B are defective. An assembled fan is found to have a defective motor. What is the probability that this motor was supplied by company B? 12

A certain transistor is manufactured at three factories at Barnsley, Bradford and Bristol. It is known that the Barnsley factory produces twice as many transistors as the Bradford one, which produces the same number as the Bristol one (during the same period). Experience also shows that 0.2% of the transistors produced at Barnsley and Bradford are faulty and so are 0.4% of those produced at Bristol. A service engineer, while maintaining electronic equipment, finds a defective transistor. What is the probability that the Bradford factory is to blame?

13

A system consists of seven components, as shown in the following diagram. Find the reliability of the system, given that the individual probabilities for functioning are: PA  0.90, PB  0.95, PC  0.95, PD  0.92, PE  0.97, PF  0.92 and PG  0.97 A

B

D

E

D F

G

C

UNIVERSITY OF TECHNOLOGY, JAMAICA SCHOOL OF MATHEMATICS AND STATISTICS 3

MODULE: ENGINEERING STATISTICS (STA2023) WORKSHEET #2 1. Consider the segment of an electric circuit with three relays shown here. Current will flow from a to b if there is at least one closed path when the switch is thrown. Each of the three relays has an equally likely chance of remaining open or closed when the switch is thrown. Let X represent the number of relays that close when the switch is thrown.

(i) (ii)

A A A B A a b A 1C A Find the probability distribution for X and display it in tabular form. A 1

What is the probability that current will flow from a to b?

(iii)

Find E(X)

(iv)

Find var(X)

2.

Turbo Generators plc manufacturer seven large turbines for a customer. Three of these turbines do not meet the customers’ specification. Quality control inspectors choose two turbines at random. Let the discrete random variable X be defined to the number of turbines inspected which meet the customer’s specification. Find the probabilities that X takes values 0, 1 or 2.

3.

In a box of switches it is known 10% of the switches are faulty. A technician is wiring 30 circuits, each of which needs one switch. What is the probability that (i)

4.

all thirty work

(ii)

at most 2 of the circuits do not work?

A University Engineering Department has introduced a new software package called SOLVIT. To save money, the University’s Purchasing department has negotiated a bargain price for a 4- user licence that allows only four students to use SOLVIT at any one time. It is estimated that this should allow 90% of students to use the package when they

4

need it. The Student’s Union has asked for more licences to be bought since engineering students report having to queue excessively to use SOLVIT. As a result the Computer Centre monitors the use of the software. Their findings show that on average 20 students are logged on at peak times and 4 of these want to use SOLVIT. Was the Purchasing Department’s estimate correct? 5.

A factory has 10 machines which may need adjustment from time to time during the day. Three of these machines are old; each having a probability of 0.1 of needing adjustment during the day, and 7 are new, having corresponding probabilities of 0.05 Assuming that no machine needs adjustment twice on the same day, determine the probabilities that on a particular day (i) (ii)

6.

Just 2 old and no new machines need adjustment. If just two machines need adjustment, they are of the same type.

Suppose the number of cracks per concrete specimen for a particular type of cement mix has approximately a Poisson distribution. Furthermore, assume that the average number of cracks per specimen is 2.5 (i) (ii) (iii)

Find the mean and standard deviation of the number of cracks per concrete specimen. Find the probability that a randomly selected concrete specimen has exactly five cracks. Find the probability that a randomly selected specimen has two or more cracks.

7.

Suppose it has been observed that, on average, 180 cars per hour pass a specified point on a particular road in the morning rush hour. Due to impending road works it is estimated that congestion will occur close to the city centre if more than 5 cars pass the point in any one minute. What is the probability of congestion occurring?

8.

Suppose vehicles arrive at a signalised road intersection at an average rate of 360 per hour and the cycle of the traffic lights is set at 40 seconds. In what percentage of cycles will the number of vehicles arriving be (a) exactly 5, (b) less than 5? If, after the lights change to green, there is time to clear only 5 vehicles before the signal changes to red again, what is the probability that waiting vehicles are not cleared in one cycle?

9.

Show that the moment-generating function for the binomial random variable is given by

M (t )   pet  q 

n

5

Use this result to derive the mean and variance for the binomial distribution. 10.

Show that the moment-generating function for the Poisson random variable with mean  is given by

M (t )  e (e 1) t

Use this result to derive the mean and variance for the Poisson distribution.

UNIVERSITY OF TECHNOLOGY, JAMAICA

6

SCHOOL OF MATHEMATICS AND STATISTICS MODULE: ENGINEERING STATISTICS (STA 2023) WORKSHEET#3 1.

2.

The mileage C in thousands which car owners get with a certain kind of tyre is a random variable having probability density function

(a)

 1  20x x0  e f ( x)   20  0 x0  Find the distribution function F(x)

(b)

Find the probability that one of these tyres will last at least 30,000 miles.

The time taken by a team to complete the assembly of an electrical component is found to be normally distributed, about a mean of 110 minutes, and with a standard deviation of 10 minutes. (a)

Out of a group of 20 teams, how many will complete the assembly: (i) Less than 95 minutes. (ii) In more than 2 hours.

(b)

If the management decides to set a “cut off” time such that 95% of the teams will have completed the assembly on time, what time limit should be set?

3.

A machining operation produces steel shafts having diameters that are normally distributed with a mean of 1.005 inches and a standard deviation of 0.01 inch. Specifications call for diameters to fall within the interval 1.00  0.02 inches. What percentage of the output of this operation will fail to meet the specifications?

4.

Suppose X has an exponential distribution density function with mean  .Show that

P( X  a  b X  a)  P( X  b) This is referred to as the ‘memoryless’ property of the exponential distribution. 5.

The life lengths of automobiles tires of a certain brand , under average driving conditions, are found to follow an exponential distribution with mean 30 (in thousands of miles).Find the probability that one these tires bought today will last (i)

Over 30,000 miles 7

(ii) 6.

Over 30,000 miles given that it already has gone 15,000 miles.

A pumping station operator observes that the demand for water at a certain hour of the day can be modelled as an exponential random variable with a mean of 100cfs (cubic feet per second). (i)

Find the probability that the demand will exceed 200cfs on a randomly selected day. What is the maximum water-producing capacity that the station should keep on line for this hour so that the demand will exceed this production capacity with a probability of only 0.01?

(ii)

7.

Let X have density function

cxe2 x f ( x)    0 (i) 8.

Find the value of c.

0 x elsewhere (ii)

Find the mean and variance of X.

Consider the gamma distribution with p.d.f given by

x   x  1   f ( x)     ( ) e  0 

x0 elsewhere

Using the moment generating function, show that (i) mean, E(X)=  ,

Var ( X )   2

(ii) 9.

The kinetic energy k associated with a mass moving at velocity v is given by the expression

k

mv 2 2

Consider a device that fires a serrated nail in to concrete at a mean velocity of 2000 feet per second, where the random velocity V possess a density function given by v

f (v ) 

v3 e 500

 500 

4

4

v0

8

Find the expected kinetic energy associated with a nail of mass m. 10.

A Weibull density function has the form x    1  x e  x0 f ( x)     0 elsewhere  The length of service time during which a certain type of thermistor produces resistances within its specifications has been observed to follow a Weibull distribution with   50 

11.

and   2 (measurements in thousands of hours). (i) Find the probability that one of these thermistors, to be installed in a system today, will function properly for over 10,000 hours. (ii) How long can such a thermistor be expected to last? 12.

Fatigue life, in hundreds of hours, for a certain type of bearings has approximately a Weibull distribution with   2 and   4 (i) (ii) (iii)

13.

Find the probability that a bearing of this type fails in less than 200 hours. Find the expected value of the fatigue life for these bearings. Find the variance of the fatigue life for these bearings.

The lifetime t (in hours) of a certain electronic component is a random variable with density function t  1 100  e f (t )  100  0 

(i) (ii)

t 0 elsewhere

Find F (t) and R (t). What is the reliability of the component at t  25 hours?

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UNIVERSITY OF TECHNOLOGY, JAMAICA SCHOOL OF MATHEMATICS AND STATISTICS MODULE: ENGINEERING STATISTICS (STA 2023) WORKSHEET#4

10

1.

Let X1 , X 2 ,........X n be a random sample of n independent observations from a population with mean  and variance  2 . Show that E ( X )   and Var ( X ) 

2.

2 n

Two-centimetre number 10 woodscrews are manufactured in their millions but packed in boxes of 200 to be used to the public or trade. If the length of the screws is known to be normally distributed with a mean of 2 cm and variance 0.05 cm2, find the mean and standard deviation of the sample mean of 200 boxed screws. What is the probability that the sample mean length of the screws in a box of 200 is greater than 2.02cm? 2 1 n 1  n 2 1 n   2 s  Yi    Yi    (Yi  Y )  n  1  n  1 i 1 n  i 1    i 1 2

3.

Show that

4.

Let Y1,Y2, ......Yn be a random sample with E (Yi ) =  and Var (Yi ) =  2 . Show that 1 n S '   (Yi  Y ) 2 is a biased estimator for  2 and that n i 1 2

S2 

5.

1 n  (Yi  Y )2 is an unbiased estimator for  2 . n  1 i 1

After 1000 hours of use the weight loss, in gm, due to wear in certain rollers in machines, is normally distributed with mean  and variance  2 .Fifty independent observations are 50

taken. If observation i is yi , then

 yi  497.2 and i 1

50

y i 1

i

2

 5473.58.

Estimate 

and  2 and give a 95% confidence interval for  6.

The fuel consumption of a new model of car is being tested. In one trial, 50 cars chosen at random, were driven under identical conditions and the distances, x km, covered on 1 litre of petrol were recorded. The results give the following totals:

 x  525,

x

2

 5625

Calculate a 95% confidence interval for the mean petrol consumption, in kilometres per litre of cars of this type. 7.

Stainless steels are frequently used in chemical plants to handle corrosive fluids. However, these steels are especially susceptible to stress corrosion cracking in certain environments. In a random sample of 300 steel alloy failures that occurred in oil refineries and

11

petrochemical plants in a country over the last 10 years, 118 were caused by stress corrosion cracking and corrosion fatigue. Construct a 95% confidence interval for the true proportion of alloy failures caused by stress corrosion cracking. 8.

To estimate the average time required for certain repairs, an automobile manufacturer had 40 mechanics, a random sample , timed in the performance of this task. If this took them on the average 24.05 minutes and a standard deviation of 2.68 minutes, find a 95% confidence interval for the mean if he uses x  24.05 minutes as an estimate of the actual mean time required to perform the given repairs?

9.

A random sample of 40 engineers was selected from among the large number employed by a corporation engaged in seeking new sources of petroleum. The hours worked in a particular week was determined for each engineer selected. These data had a mean of 46 hours and a standard deviation of 3 hours. For that particular week, estimate the mean hours worked for all engineers in the corporation using a 95% confidence interval.

10.

The breaking strength of threads has a standard deviation of 18 grams. How many measures on breaking strength should be used in the next experiment if the estimate of the mean breaking strength is to be within 4 grams of the true mean breaking strength, with confident coefficient 0.95?

11.

Vehicle speed needs to be estimated with an accuracy of 5 mph of the average speed with a 99% confidence interval. Assume that the standard deviation of the vehicle speed is 10mph. How many samples are required?

12.

Upon testing 100 resistors manufactured by Company A, it is found that 12 fail to meet the tolerance specifications. (i)

Find 95% confidence interval for the true fraction of resistors manufactured by Company A that fail to meet the tolerance specification.

(ii)

If it is desired to estimate the true proportion failing to meet tolerance specifications to within 0.05, with confidence coefficient 0.95, how many resistors should be tested?

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UNIVERSITY OF TECHNOLOGY-JAMAICA SCHOOL OF ENGINEERING BEng2 Engineering Statistics

STA2023 Tutorial Sheet#3A

Question#1 a)

A random variable X has the probability density function ce  x . Find the proper value of c, assuming 0  X  . Find the mean and variance of X.

b)

The cumulative distribution function that a television tube will fail in t-hours is 1  e ct , where c is a parameter dependent on the manufacturer and t  0. Find the p.d.f. of T, the life of the tube.

Question#2

13

a)

The manager of a men’s clothing store is concerned over the inventory of suits, which is currently 30 (all sizes). The number of suits sold from now to the end of the season is distributed as:  e 20 20 x ,  P  X  x    x!  0, 

x  0,1, 2,3,....... otherwise

Find the probability that he would have suits left over at the season’s end. b)

A random variable X has a CDF of the form: x 1  1  1   FX  x    , 2 0, otherwise 

x  0,1, 2,3,............

i)

Find the probability function for X

ii)

Find P  0  X  8 .

Question#3 a)

Consider the following probability density function:

 kx, 0  x < 2  f  x   k  4  x  , 2  x  4  0, otherwise  Find: i)

k

ii)

mean and variance of X

iii)

the CDF of X.

b)

Find the cumulative distribution function CDF associated with:

i)

 x  2xt22 , t  0, x 0  e f ( x)   t 2  0, otherwise 

14

f  x 

ii)

1

1

 

  x   2     1   2       

,

- < x < 

Question#4 a)

A random variable X has the following probability density function:

 e x , x > 0. f  x   0, otherwise Develop the density function for:

b)

i)

Y  2X 2

ii)

V X2

iii)

U  ln X .

1

A random variable X has the following probability density function: ax 2 e  bx , x > 0. f  x    0, otherwise 2

i) ii)

Find a. Suppose a new function Y  18 X 2 is of interest. Find an approximate value for

E  X  and V  X  . Question#5 a)

A contractor is going to bid on a project and the number of days, X, required for completion follows the probability distribution given as:

    P  X  x       The contactor’s profit is

0.1, x  10 0.3, x  11 0.4, x  12 0.1, x  13 0.1, x  14 0, otherwise

Y  2000 12  X  .

15

i)

Find the probability distribution of Y.

ii)

Find E  X  ,V  X  , E Y  , andV Y  .

UNIVERSITY OF TECHNOLOGY-JAMAICA SCHOOL OF ENGINEERING

SUBJECT: ENGINEERING STATISTICS

Tutorial Sheet#3B

Question#1

A manufacturing process produces certain articles such that the probability of each article being defective is p. Let Y be the r.v. denoting the minimum number of articles to be manufactured, until the first two defective articles appear.

i)

Show that the distribution of Y is given by

P(Y  y)  p 2 ( y  1)(1  p) y 2 ,

ii)

y  2,3, 4,.....;

Find E ( X ) and V ( X )

16

iii)

Calculate the probability P(Y  100) for p  0.05.

Question#2

a)

The discrete random variable X has a logarithmic series distribution with

P( X  x) 

 qx

, x  1, 2,3,............ x where 0  q  1  p  1.

Show that:

1 ln p

i)



ii)

E( X ) 

q 1 q

iii)

V (X ) 

17

 q(1   q) (1   )2

Question#3

a)

b)

Prove the following results

i)

 n   n  1   n  1      r   r   r  1

ii)

 m  n  m  m  n          k  r 0  r  k  r 

Consider the hyper-geometric discrete random variable X, which has probability distribution:

 M  N  M     x  n  x   , where x  n, x  M , and n  x  N  M . P( X  x)  N   n Show that:

M   N

i)

  n

ii)

 2  n    N 

 M  N  M  N  n   . N  N  1 

Question#4

18

a)

The decapitated geometric r.v. is a particular case of the shifted geometric variable X with:

P( X  x)  q x 3 p, x  3, 4,5,.........

Determine:

b)

i)

the moment generating function M X (t ) of X.

ii)

E ( X ) and V ( X ).

Let X has a zero-truncated (or decapitated) Poisson distribution with zero class missing, i.e.:

P( X  x) 

(em  1)1 k x , x  1, 2,3,..... x!

i)

Find the moment generating function , M X (t )

ii)

Find E ( X ). and V ( X ).

19

Question#5

a)

A random variable X has a probability density given by:

2 xe  x for x  0 f ( x)   elsewhere  0 2

where   0. Show that for this distribution:

i)

ii)

b)

Let X



1  2  

1     1    4



t Note that: e dt 

2

0

B(n, p) , use the moment generating function technique to find:

i)

M X (t )

ii)

E( X )

iii)

V ( X ).

Question#6

a)

2

A continuous random variable X having p.d.f.

20

 2

.

f ( x)   e (  ) x , x  

is said to be an exponential random variable with threshold

.

Find:

b)

i)

the moment generating function, M X (t ), of X.

ii)

E ( X ) and V ( X ).

The velocities of gas particles can be modeled by the Maxwell distribution, with the probability density function given by:

 m  f (v)  4    2 KT 

3/ 2

v 2e v

2

( m / 2 KT )

, v0

where m is the mass of the particle, K is Boltzmann’s constant, and T is the absolute temperature.

Find the mean velocity of these particles.

c)

Let X be a uniform (rectangular) random variable which is distributed on the interval ( ,  ) .

21

Show that its m.g.f. M X (t ) is given by:

et   et M X (t )  t (   )

Hence, find E ( X )

and V ( X ).

Question#7

a)

A continuous random variable X is said to have a generalized (3-parameter) gamma distribution if its p.d.f. is

f ( x) 

c c  ck x ck 1e (  x ) , 0  x  , (c  0, k  0,   0) ( k )

Find:

i)

b)

E(X)

ii)

V(X)

A random variable X has a Weibull distribution if and only if its probability density is given by

mx  1e  x for x  0 f ( x)   elsewhere  0 

where   0 and   0.

22

c)

i)

Express m in terms of  and  .

ii)

Show that    1/   1 

Let X





1  

Geo( p ) , use the moment generating and the probability generating function

technique, to find:

i)

M X (t )

ii)

E( X )

iii)

V ( X ).

Question#8

a)

A random variable X is said to have a Cauchy distribution if its density is given by:

  f ( x)  , for    x   [ x   ]2   2

Find:

E ( X ) and V ( X ) if they exist.

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b)

A random variable X has a Pareto distribution if and only if its probability density is given by:

  , for x  1  f ( x)   x 1 0, elsewhere where   0.

 , provided   1.  1

i)

Show that for the Pareto distribution  

ii)

Find, also V ( X ) and the restrictions on  for the existence of V ( X ).

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UNIVERSITY OF TECHNOLOGY, JAMAICA SCHOOL OF MATHEMATICS AND STATISTICS MODULE: ENGINEERING STATISTICS (STA 2023) WORKSHEET # 5 1.

The output voltage for a certain circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean of 128.6 and a standard deviation of 2.1. Test the hypothesis that the average output voltage is 130 against the alternative hypothesis that it is less than 130. Use a 5% significance level.

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2.

A machine is set to produce bolts with a mean length of 1 inch. Bolts that are too long or too short do not meet customer’s specifications and must be rejected. To avoid producing too many rejects, the bolt produced by the machine are sampled from time to time and tested as a check to determine whether the machine is still operating properly, i.e., producing bolts with a mean length of 1 inch. Suppose 50 bolts have been sampled, and

x  1.02 inches and s = 0.04 inch. Does the sample evidence indicate that the machine is producing bolts with a mean length not equal to 1 inch; that is the production is out of control? Use a 0.01 level of significance. 3.

A firm manufactures heavy current switch units which depend for their correct operation on a relay. The relays are provided by an outside supplier and out of a random sample of 150 relays delivered, 140 are found to work correctly. Can the relay manufacturer justifiably claim that at least 90% of the relays provided will function correctly? Test at the 0.05 level of significance.

4. SwitchRight, a manufacturer of engine management systems requires its supplier of control modules to supply modules with at least 99% complying with their specification. The quality control operators at SwitchRight check a random sample of 1000 control modules delivered to SwitchRight and find that 985 match the specification. Does this result imply that less than 99% of the control modules supplied do not match Switch Right specification? Test at the 0.05 level of significance.

5. The means and standard deviations shown in the table summarize information on the strengths for two types of wooden poles used by the utility industry. Do the data provide sufficient evidence to indicate a difference in the mean of the strengths of wooden poles made from coastal Douglas fir and southern pine? Use a 0.05 level of significance.

Species

Sample size

Sample mean

Standard deviation

Coastal Douglas fir

118

8380

644.62

Southern pine

147

8870

611.72

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6.

A motor manufacturer wishes to replace steel suspension components by aluminium components to save weight and thereby improve performance and fuel consumption. Tensile strength tests are carried out on randomly chosen samples of two possible components before a final choice is made. The results are: Component Number 1

Sample size 35

2

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MeanTensile Strength (kg mm-2) 90 88

Standard deviation (kg mm-2) 2.3 2.2

Is there any difference between the measured tensile strengths at the 5% level of significance? 7.

Scientists have labelled benzene, a chemical solvent commonly used to synthesize plastics, as a possible cancer-causing agent. Studies have shown that people who work with benzene more than 5 years have 20 times the incidence of leukemia than the general population. Suppose a steel manufacturing plant , which exposes its workers to benzene daily, is under investigation by the Occupational Safety and Health Administration. 20 air samples, collected over a period of 1 month and examined for benzene content, yielded the following summary statistics: s  1.7 ppm Is the steel manufacturing plant in x  2.1 ppm violation of the new government standards? Test the hypothesis that the mean level of benzene at the steel manufacturing plant is greater than 1 ppm. Use a 0.05 level of significance.

8.

An important measure of the performance of a machine is the mean time between failures (MTBF). A certain printer attached to a processor was observed for a period of time during which 10 failures were observed. The times between failures averaged 98 working hours with a standard deviation of 6 hours. A modified version of this sprinter was observed for 8 failures, the times between failures averaging 94 working hours with a standard deviation of 8 hours. Can we say, at the 1% significance level, that the modified version of the printer has a smaller MTBF?

9.

A manufacturer of electronic equipment has developed a circuit to feed current to a particular component in a computer display screen. While the new design is cheaper to manufacture, it can only be adopted for mass production if it passes the same average current to the component. In tests involving the two circuits, the following results are obtained.

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Test Number 1 2 3 4 5 6 7 8 9 10 11 12

Circuit1 – Current (mA) 80.1 82.3 84.1 82.6 85.3 81.3 83.2 81.7 82.2 81.4

Circuit 2 – Current (mA) 80.7 81.3 84.6 81.7 86.3 84.3 83.7 84.7 82.8 84.4 85.2 84.9

X 1  82.42 X 2  83.72 S12  2.00 S22  2.72 On the assumption that the populations from which the samples are drawn have equal variances, should the manufacturer replace the old circuit design by the new one? Use the 0.05 level of significance. 10.

In an experiment to determine the most advantageous position in a machine to mount an electric component which may be prone to failure due to excessive heat build-up, 300 machines are tested with 100 randomly chosen examples of the component in each of 3 positions. The results obtained were as follows. Position Failure Non-failure Total

1 40 60 100

2 30 70 100

3 50 50 100

Total 120 180 300

Use a  2 test at the 0.05 level of significance to determine whether component failure is related to mounting position.

UNIVERSITY OF TECHNOLOGY, JAMAICA SCHOOL OF MATHEMATICS AND STATISTICS ENGINEERING STATISTICS (STA2023/)

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WORKSHEET# 6 1.

The data in the following table give the miles per gallon obtained by a test automobile when using gasolines of varying octane levels. Miles per gallon(y) Octane(x)

13.0

13.2

13.0

13.6

13.3

13.8

14.1

14.0

89

93

87

90

89

95

100

98

(i) (ii) (iii) (iv) (v) (vi)

2.

3.

Draw a scatter diagram for this data. Find the least square regression equation for data. Predict the miles per gallon when then octane level is 92. Calculate the coefficient of correlation , r .Interpret this value. Calculate the coefficient of determination, r 2 . Interpret this value. Do the data provide sufficient evidence to indicate that octane level and miles per gallon are dependent? Give the attained significance level, and indicate your conclusion if you wish to implement an   0.05 level test. The following data give the diffusion time (hours) of a silicon wafer used in manufacturing integrated circuits and the resulting sheet resistance of transfer. Diffusion time(x)

0.56

1.10

1.58

2.00

2.45

Sheet resistance (y)

83.7

90.0

90.2

92.4

91.6

(i)

Find the equation of the least squares line fit to these data.

(ii)

Predict the sheet resistance when the diffusion time is 1.3hours

The quality control engineer at Bethel steel is interested in estimating the tensile strength of steel wire based on its outside diameter and the amount of molybdenum in the steel. As an experiment, she selected twenty – five pieces of wire, measured the outside diameters, and determined the molybdenum content. Then she measured the tensile strength of each piece. The results of the first four were:

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Piece A B C D

Tensile Strength (psi) Outside Y X1 11 9 16 12

Diameter(mm) Amount of Molybdenum(units), X 2 0.3 6 0.2 5 0.4 8 0.3 7

Show that the multiple regression equation is Y '  0.5  20 X 1  1X 2 (i) (ii) (iii)

Based on the equation, what is the predicted tensile strength of a steel wire having an outside diameter of 0.35 mm and 6.4 units of molybdenum? Interpret the value of b1 in the equation Suppose that over an extended past period of time/ the slope of the

 relationship between Y and X1 was 20.32. The standard error of the coefficient for b1, s b1 , is 0.12. Test at the 0.01 level of significance if there has been a change in the value of b1.

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