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UNIVERSITY OF KWAZULU-NATALSCHOOL OF MATHEMATICS, STATISTICS& COMPUTER SCIENCEMAIN EXAMINATIONJUNE 2015COURSE AND CODEEngineering Statistics STAT370 HDURATION: 2 Hours TOTAL MARKS: 32INTERNAL EXAMINER: Ms. D. RobertsEXTERNAL EXAMINER: Mr. J. Hammujuddy, University of KwaZulu-NatalTHIS EXAM C...

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UNIVERSITY OF KWAZULU-NATAL SCHOOL OF MATHEMATICS, STATISTICS & COMPUTER SCIENCE MAIN EXAMINATION JUNE 2015 COURSE AND CODE Engineering Statistics STAT370 H1

DURATION: 2 Hours

TOTAL MARKS: 32

INTERNAL EXAMINER: Ms. D. Roberts EXTERNAL EXAMINER: Mr. J. Hammujuddy, University of KwaZulu-Natal THIS EXAM CONSISTS OF 32 QUESTIONS AND 19 PAGES INCLUDING THIS ONE. PLEASE ENSURE THAT YOU HAVE ALL PAGES. INSTRUCTIONS:        

Please fill in your surname, initials and student number and seat number on the MCQ answer sheet. Please use an HB pencil and shade in clearly. Only round off at your final answer. Calculators are allowed. Answer ALL questions as there is NO negative marking. There are extra pages at the back of your question paper for rough working as well as a formula sheet, these may be detached. You will be supplied with a set of statistical tables. Do not write on them. They will be collected at the end of the exam. Each question carries 1 mark.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 1 A research council is concerned that residents of a particular town may be ingesting small but harmful amounts of lead introduced into their drinking water. Samples of drinking water taken from various locations in this town resulted in the following stem-and-leaf plot of lead content (in milligrams per 100 litres): Stem Leaf 2 015 3 478899 4 1127789 5 001455 6 667 7 0112 8 146 Key: 4|1 = 4.1 What is the range of the amount of lead in these samples? a) b) c) d) e)

6.1 6 6.6 6.131 66

Question 2 There is a group of 20 friends who applied to study Engineering at UKZN. However, there are only three places left in the Engineering Department. How many possible ways can the three students be selected to study at UKZN such that one student is selected to study Mechanical Engineering, one is selected to study Civil Engineering and the other is selected to study Electronic Engineering? a) b) c) d) e)

27 1140 60 8000 6840

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 3 How many distinct arrangements can be made from the letters of the word ENGINEERING if the arrangement must start with an E and end with a G? a) b) c) d) e)

15 120 362 880 1 663 200 277 200 2 520

Question 4 The probability that a data-communications system will have high selectivity is 0.78 and the probability that it will have high fidelity is 0.64. The events of this system having high selectivity and high fidelity are independent of each other. What is the probability this system will have neither high selectivity nor high fidelity? a) b) c) d) e)

0.5008 0.0792 0.4200 0.4992 There is not enough information for this to be calculated.

Question 5 A firm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas 92% of the time. New operators who do not take the training course only meet their quotas 64% of the time. It is known that 75% of new operators attend the training course. If a new operator meets their production quota, what is the probability that they attended the training course? a) b) c) d) e)

0.2267 0.3125 0.6900 0.0824 0.8118

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 6 Let X be the number of defects found on a solar heating panel developed by a particular manufacturer. The probability distribution of X is as given in the table below. It is known that the expected number of defects on a solar heating panel is 1.29. x

0

1

2

3

P(x)

a

0.37

b

0.12

The values of a and b are, respectively, a) b) c) d) e)

0.49 and 0.05 0 and 0.80 0 and 0.51 0.23 and 0.28 0.38 and 0.13

Questions 7 to 9 are based on the following information: When particles are released into an evacuated duct and collide with the inner duct wall, they can either be absorbed by the wall or reflected off it. Each particle can be independently absorbed with a probability of 0.4. Suppose 12 particles are released into the duct. Question 7 What is the probability that 7 particles are absorbed by the wall? a) b) c) d) e)

0.0363 0.0001 0.0959 0.0133 0.1009

Question 8 What is the probability that more than 5 particles are absorbed by the wall? a) b) c) d) e)

0.5618 0.8416 0.3348 0.6652 0.2270

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 9 What is the expected number of particles that are reflected by the wall? a) b) c) d) e)

7.2 4.8 3 4 2.9

Question 10 A hardware store has a total of 30 washers to be sold. Seven of the washers have circumferences too thick for them to be used. If a customer buys a batch of 10 washers, what is the probability that they bought at least 2 washers that cannot be used? a) b) c) d) e)

0.6959 0.7715 0.2135 0.8096 0.4288

Questions 11 and 12 are based on the following information: A manufacturer produces a very large amount of copper piping and therefore these pipes are often prone to defects. There are, on average, 2 dents per metre of copper piping produced by this manufacturer. Question 11 What is the probability that there are 5 dents on three metres of copper piping? a) b) c) d) e)

0.1606 0.4366 0.0361 0.8773 0.1083

P.T.O.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 12 What is the standard deviation of the number of dents found on 50 metres of coping piping produced by this manufacturer? a) b) c) d) e)

1.4 100 2 3.2 10

Question 13 A new cleaning device has been developed to help reduce the amount of pollution discharged into the ocean. Let 𝑋 denote the amount of pollutants (parts per million) removed when the device is operating. The cumulative distribution function of 𝑋 is given below: 0 0.2𝑥 𝐹(𝑥) = 0.04𝑥 + 0.64 { 1

𝑓𝑜𝑟 𝑥 < 0 𝑓𝑜𝑟 0 ≤ 𝑥 < 4 𝑓𝑜𝑟 4 ≤ 𝑥 < 9 𝑓𝑜𝑟 𝑥 ≥ 9

What is the probability that more than 2 but at most 7 pollutants are removed when this device is operating? a) b) c) d) e)

0.32 0.20 0.44 0.52 0.92

P.T.O.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Questions 14 to 16 are based on the following information: The reaction time of a motorbike rider to visual stimulus is normally distributed with a mean of 0.30 seconds and a standard deviation of 0.06 seconds. Question 14 What is the probability that a reaction of a motorbike rider requires more than 0.40 seconds? a) b) c) d) e)

0.0465 0.6554 0.0475 0.3446 0.9525

Question 15 If 79.67% of motorbike riders take longer than k seconds to react, what is the value of k? (Round your final answer off to two decimal places) a) b) c) d) e)

0.35 0.25 0.83 0.20 0.19

Question 16 If a sample of 20 motorbike riders is taken, what is the probability that the average of their reaction time is less than 0.34 seconds? a) b) c) d) e)

0.6667 0.9981 0.7486 0.9986 0.0014

P.T.O.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Questions 17 to 20 are based on the following information: Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic and specifications require that the mean burning rate must be 50 centimetres per second. It is known that the standard deviation of burning rate is 2 centimetres per second. In order to test if these specifications are being met, a hypothesis test is carried out using a 5% level of significance and the following hypotheses: 𝐻0 : 𝜇 = 50 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇 ≠ 50

A sample of 20 propellants resulted in a mean burning rate of 51 centimetres per second. Question 17 When will the null hypothesis be rejected? a) b) c) d) e)

If the test statistic is less than -1.96 or greater than 1.96 If the test statistic is less than -1.645 or greater than 1.645 If the test statistic is greater than 1.645 If the test statistic is greater than 2.24 If the test statistic is less than -2.093 or greater than 2.093

Question 18 What is the p-value for the above test? a) b) c) d) e)

0.0125 0.0500 0.0250 0.9875 0.0038

Question 19 Suppose the null hypothesis in the above test is rejected, what conclusion can be made? a) There is enough information to conclude that the mean burning rate is not 51 centimetres per second. b) There is enough information to conclude that the mean burning rate is 50 centimetres per second. c) There is enough information to conclude that the mean burning rate is more than 50 centimetres per second. d) There is enough information to conclude that the mean burning rate is less than 50 centimetres per second. e) There is enough information to conclude that the mean burning rate is not 50 centimetres per second.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 20 What sample size should be taken if the mean burning rate is to be estimated with 99% confidence and an error of no more than 0.5 centimetres per second? a) b) c) d) e)

11 107 87 106 86

Questions 21 to 23 are based on the following information: Two suppliers manufacture a plastic gear used in a laser printer. The higher the impact strength of these gears (measured in foot-pounds), the longer the laser printer will last. The standard deviation of the impact strength of the gears produced by both suppliers is known to be the same. The results from a random sample of gears from each supplier is summarized below:

Sample Size

Mean

Standard Deviation

Supplier 1 (Population 1)

18

254

12

Supplier 2 (Population 2)

21

314

24

Supplier 2 claims that their laser printers last longer than those from Supplier 1. Question 21 If the claim of Supplier 2 is to be tested at a 10% level of significance, which of the following represents the null and alternate hypotheses? a) 𝐻0 : 𝜇1 > 𝜇2 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇1 < 𝜇2

b) 𝐻0 : 𝜇1 ≠ 𝜇2 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇1 = 𝜇2

c) 𝐻0 : 𝜇1 = 𝜇2 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇1 ≠ 𝜇2 d) 𝐻0 : 𝜇1 = 𝜇2 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇1 < 𝜇2

e) 𝐻0 : 𝜇1 = 𝜇2 𝑣𝑒𝑟𝑠𝑢𝑠 𝐻1 : 𝜇1 > 𝜇2

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 22 The test statistic for the above test is a) b) c) d) e)

-9.61 -10.08 -0.49 -1.31 -43.44

Question 23 Suppose a 98% confidence interval for the difference in the mean strength of the gears made by Supplier 1 and Supplier 2 is (-75.17 ; -44.83), which means a) We are 98% confident that the mean strength of the gears made by Supplier 1 is between 44.83 and 75.17 units more than the mean strength of the gears made by Supplier 2. b) 98% of the time the mean strength of the gears made by Supplier 1 is between 44.83 and 75.17 units less than the mean strength of the gears made by Supplier 2. c) 98% of the time the mean strength of the gears made by Supplier 1 is between 44.83 and 75.17 units more than the mean strength of the gears made by Supplier 2. d) We are 98% confident that the mean strength of the gears made by Supplier 1 is between 44.83 and 75.17 units less than the mean strength of the gears made by Supplier 2. e) There is a probability of 0.98 that the mean strength of the gears made by Supplier 1 is between 44.83 and 75.17 units less than the mean strength of the gears made by Supplier 2.

Questions 24 and 25 are based on the following information: The melting points (in degrees Celsius) of two alloys used in formulating solder were investigated by melting samples of each material. The results of the samples are summarized below: Sample Size

Mean

Variance

Alloy 1 (Population 1)

15

215

49

Alloy 2 (Population 2)

32

258

36

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 24 When testing whether or not the population variances of the melting points for Alloy 1 and Alloy 2 are equal, a type I error will be made if a) 𝐻0 : 𝜎12 = 𝜎22 is true but not rejected

b) 𝐻0 : 𝜎12 ≠ 𝜎22 is true but rejected.

c) 𝐻0 : 𝜎12 = 𝜎22 is true but rejected.

d) 𝐻0 : 𝜎12 = 𝜎22 is false but not rejected. e) 𝐻0 : 𝜎12 = 𝜎22 is false but rejected.

Question 25 Suppose the null hypothesis for the test of equal population variances is rejected. Which one of the following represents the lower confidence limit of a 95% confidence interval for the difference in the mean melting points for Alloy 1 and Alloy 2? a) b) c) d) e)

-72.23 -48.11 -46.59 -46.26 -47.33

Question 26 Consider the following confidence interval for the difference between two population means when the population variances are known: (𝑥1 − 𝑥2 ) ± 𝑧1−𝛼⁄2√

𝜎12 𝜎22 + 𝑛1 𝑛2

If both the sample sizes increase while all the other values stay the same, the confidence interval will a) b) c) d) e)

stay the same width but shift left. become narrower. shift to the left and get wider. shift to the right but stay the same width. become wider.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Questions 27 and 28 are based on the following information: An experiment was run to determine whether four specific firing temperatures affect the density of a certain type of brick. To test whether the mean densities for all the firing temperatures are equal, six densities were measured for each temperature. A partially completed ANOVA table is shown below. Source Temperature

Sum of squares

df

Mean square

F-statistic

30.58

(ii)

(iii)

8.22

Error

(i)

Total

55.43

1.24

Question 27 The entries in the ANOVA table labeled (i), (ii) and (iii) have values (correct to two decimal places) a) b) c) d) e)

(i) 24.85 (i) 24.85 (i) 24.85 (i) 29.76 (i) 29.76

(ii) 4 (ii) 20 (ii) 3 (ii) 3 (ii) 3

(iii) 7.65 (iii) 1.53 (iii) 10.19 (iii) 6.63 (iii) 10.19

Question 28 When testing the hypothesis of equal mean densities for the different firing temperatures at a 5% level of significance, the decision and conclusion will be a) b) c) d) e)

Do not reject 𝐻0 . There is enough evidence to conclude at least two means differ. Reject 𝐻0 . There is enough evidence to conclude all the means differ. Reject 𝐻0 . There is enough evidence to conclude all the means are the same. Reject 𝐻0 . There is enough evidence to conclude at least two means differ. Do not reject 𝐻0 . There is not enough evidence to conclude at least two means differ.

P.T.O.

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 29 A mechanical engineer is studying the surface roughness of a part produced in a metal-cutting operation. Two factors on the effect of surface roughness are of interest; depth of cut and tool angle. Two measurements for each combination of these factors were taken and the results shown in the table below: Tool Angle (Factor B) 15°

25°

0.025 inches

9

7

10

11

0.040 inches

11

12

12

15

Depth of Cut (Factor A)

Determine the estimated average effect of the interaction between the depth of cut and tool angle a) b) c) d) e)

-0.85 -0.25 21.75 10.88 -0.13

Questions 30 to 32 are based on the following information: Consider the table below which shows the purity of oxygen produced (𝑦) in a chemical distillation process according to the percentage of hydrocarbons (𝑥 ) that are present in the main condenser of the distillation unit. Hydrocarbons (%)

0.99

1.02

1.15

1.29

1.46

1.23

1.55

1.40

1.36

0.87

Purity (%)

90.01

89.05

91.43

93.74 96.73

91.77

99.42

93.65

94.45

87.89

∑ 𝑥 = 12.32,

∑ 𝑦 = 928.14,

∑ 𝑥𝑦 = 1150.2402, ∑ 𝑥 2 = 15.6206,

∑ 𝑦 2 = 86256.8744

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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Question 30 The gradient of the least squares regression line is a) b) c) d) e)

0.06 73.95 0.96 17.34 15.31

Question 31

Suppose 𝑏0 = 73.95 and 𝑆𝑒 = 1.05, which of the following represents the test statistic when testing if the true intercept of the line is 75? (Round your final answer off to four decimal places) a) b) c) d) e)

-0.5765 -0.6651 -0.5322 -30.2518 -0.2715

Question 32 The coefficient of correlation between the purity of oxygen produced in a chemical distillation process and the percentage of hydrocarbons present is a) b) c) d) e)

0.96 0.23 0.14 0.06 0.98

The End 

University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

P.T.O. for rough working paper and formula sheet. (These may be detached)

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University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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University of KwaZulu-Natal, June 2015 Examination: STAT370 H1

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