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Practice Problems

Question 01

20 clouds were seeded with silver nitrate and the acre-feet of rain that resulted were recorded.

The 20 acre-feet values were

18.0 30.7 19.8 27.1 22.3 18.8 31.8 23.4 21.2 27.9

31.9 27.1 25.0 24.7 26.9 21.8 29.2 34.8 26.7 31.6

(a)

Using a P-value method can you support a claim that mean rainfall from seeded clouds exceeds

25 acre-feet ? = 0.05

Do you use a Z test or t test and why ?

What is H0 in words and symbols ? What is H1 in words and symbols ?

Select DO NOT REJECT H0 or REJECT H0.

Write CONCLUSION in words.

(b)

Show how you can reach the same CONCLUSION using a Confidence Interval method

Question 01 Solution (a) and (b)

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Question 02

A melting point test of a n=10 sample of a binder used in making rocket propellant resulted in

mean Xbar=154.2 degF. Assume that the melting point is normally distributed with standard

deviation = 1.5 degF.

(a)

Test H0:mu=155 versus H1:mu155 using =0.01

Do you use Z test or t test and why ?

Write your CONCLUSION

(b)

If n=30 (and Xbar and do not change) does your conclusion in (a) change ?

If so, explain.

Question 02 Solution

(a) and (b) calculations

(b)

Changing from n=10 to n=30 changes Standard Error of Mean, Z value and P value enough to

change CONCLUSION

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Question 03

Tensile strength tests were performed on two different grades of aluminium spars used in aircraft

wings. From past experience the standard deviations 1 2 of tensile strengths are known.

Data is as follows

Aluminium grade Grade1 Grade2

Sample size 10 12

Sample Xbar 87.6 74.5

Known 1.00 1.50

If population means 1 and 2 denote the true mean tensile strengths of the two grades, find the

90% confidence interval for the difference in the mean strength 1 - 2.

Question 03 Solution

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Question 04 Simple Linear Regression

Given the following data on Hydrocarbon levels x% and Oxygen Purity Levels y%

(a) Do a scatter plot of the data

(b) find the linear relationship between the two variables.

(c) find the coefficient of determination r2. What does it tell you ?

(d) find the correlation coefficient r. Is it positive or negative ? Why ?

Observation Hydrocarbon x% Oxygen Purity y%

1 0.99 90.01

2 1.02 89.05

3 1.15 91.43

4 1.29 93.74

5 1.46 96.73

6 1.36 94.45

7 0.87 87.59

8 1.23 91.77

9 1.55 99.42

10 1.40 93.65

11 1.19 93.54

12 1.15 92.52

Question 04 Solution

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Question 05 Simple Linear Regression

Given the following data on the fretting wear of mild steel y (10-4 cubic millimetres) and oil

viscosity x

(a) Do a scatter plot of the data. Is simple linear regression plausible ?

(b) find the linear relationship between the two variables.

(c) find the coefficient of determination r squared. What does it tell you ?

(d) Predict the fretting wear y when viscosity x=30

(e) Obtain the fitted value of y when x=22.0 and calculate the residual

Observation Viscosity x Fretting Wear y

1 1.6 240

2 9.4 181

3 15.5 193

4 20.2 155

5 22.0 172

6 35.5 110

7 43.0 113

8 40.5 75

9 33.0 94

Question 05 Solution

Q 05 solution continues next page

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Q 05 solution continued

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Question 06 Chi Square Goodness of Fit

A company operates four machines three shifts each day. Production records showing the

number of breakdowns are

Machine

Shift A B C D

1 41 20 12 16

2 31 11 9 14

3 15 17 16 10

Test the hypothesis that 1/3 of breakdowns occur on shift 1, 1/3 on shift 2 and 1/3 on shift 3

Question 06 Solution

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Question 07 Chi Square Independence of Variables

A study of four types of failures of an electronic component and the two mounting positions of the

component gave the number of failures in the table below.

Would you conclude the type of failure is independent of the mounting position ?

Use =0.01. Find P value for the test.

Failure Type

Mounting Position A B C D

1 22 46 18 9

2 4 17 6 12

Question 07 Solution

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Question 8

There are three sources of power, Electricity (E), Gas (G) and Oil (O).

The probabilities of shortages in the coming winter are

P(E shortage) = P(E) = 0.15, P(G shortage) = P(G) = 0.1, P(O shortage) = P(O) = 0.2

Use an Event Tree to calculate the probabilities

(a) There will a shortage of all three energy sources

(b) There will be a shortage in at least one of the following : Gas, Electricity

Question 8 Solution

Event Tree with Probabilities

P(Elec)=0.15

1-P(Gas)=0.90

1-P(Oil)=0.80

1-P(Elec)=0.85

P(Oil)=0.20

1-P(Oil)=0.80

P(Elec)=0.15

1-P(Elec)=0.85

P(Elec)=0.15

1-P(Elec)=0.85

P(Elec)=0.15

1-P(Elec)=0.85

P(G & O & E)

0.1*0.2*0.15=0.003

P(G & O & not E)

0.1*0.2*0.85=0.017

P(G & not O & E)

0.1*0.8*0.15=0.012

P(G & not O & not E)

0.1*0.8*0.85=0.068

P(not G & O & E)

0.9*0.2*0.15=0.027

P(not G & O & not E)

0.9*0.2*0.85=0.153

P(not G & not O & E)

0.9*0.8*0.15=0.108

P(not G & not O & not E)

0.9*0.8*0.85=0.612

P(Oil)=0.20

P(Gas)=0.1

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Sum of Probabilities in RHS boxes

=0.003+0.017+0.012+0.068+0.027+0.153+0.108+0.612 = 1.000

Question 1 Answers (a) Probabilty(Gas shortage & Oil shortage and Electricity shortage) = 0.1*0.2*0.15 =

= 0.003

(b) Probability(Gas shortage OR Electricity shortage OR Both)

= 0.003+0.017+0.012+0.068+0.027+0.108 = 0.235

Question 9

A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all

invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at

random. What is the probability that fewer than 2 of the 12 sampled invoices receive the

discount?

Question 9 Solution

Modelled as binomial distribution

Probability that receive discount(p) = 0.1

Probability that fewer than 2 invoices receive discounts out of 12 samples = P (X 1)

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CUMULATIVE AREA UNDER NORMAL CURVE (Z Table)

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CUMULATIVE AREA UNDER NORMAL CURVE (Z Table)

Continued

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t DISTRIBUTION

Area shown in 1 Tail (the Upper Tail)

Area shown in 2 Tails. / 2 in each tail

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One Tail t Values for and Degrees of Freedom

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Two Tail t Values for and Degrees of Freedom

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2 DISTRIBUTION

CRITICAL VALUES 2 for Upper Tail Area

for given Degrees of Freedom df