BTEC HNC - Analytical Methods - Statistics and Probability

Statistics and ProbabilityAnalytical Methods for Engineers

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsStatistics and Probability

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 4 -

1.1 The diameter in millimetres of a reel of wire is measured in 48 places and the results are as shown. - 4 -

(a) Form a frequency distribution of diameters having 6 classes. - 4 -Solution:- - 4 -

(b) Draw a histogram depicting the data. - 5 -Solution:- - 5 -

(c) Form a cumulative frequency distribution. - 6 -Solution:- - 6 -

(d) Draw an ogive for the data. - 6 -Solution:- - 6 -

TASK 2 - 7 -

2.1 Determine the mean, median and mode values for the following set of data correct to 1 decimal place: - 7 -

Solution:- - 7 -

2.2 The values of capacitance in microfarads selected at random from a large batch of similar capacitors: - 8 -

Solution:- - 8 -

TASK 3 - 9 -

3.1 In an experiment to determine the relationship between the current flowing in an electrical and the applied voltage, the results obtained are:- - 9 -

Solution:- - 9 -

3.2 The relationship between the voltage applied to an electrical circuit and the current flowing is as shown: - 11 -

Solution:- - 11 -Check:- - 13 -

2


TASK 4 - 14 -

4.1 A box of fuses are all of the same shape and size and comprises 23 off 2A fuses, 47 off 5 A fuses and 69 off 13 A fuses. Determine the probability of selecting at random :- - 14 -

(a) A 2 A fuse - 14 -

(b) A 5 A fuse - 14 -

(c) A 13 A fuse. - 14 -Solution (a):- - 14 -Solution (b):- - 14 -Solution (c):- - 14 -Check:- that the sum of a, b and c is equal to 1: - 14 -

4.2 An automatic machine produces on average 10 % of its components outside of the tolerance required. In a sample of 10 components from this machine determine the probability of having three components outside of the tolerance required by assuming a Binomial distribution. - 15 -

Solution:- - 15 -

4.3 In problem 4.2 determine the probability of having three components outside of the required tolerance using the Poisson distribution. - 16 -

Solution:- - 16 -Check:- - 16 -

4.4 The specific resistance of a reel of silver wire of nominal diameter 0.5 mm is estimated by determining the resistance of 7 samples of the wire. These were found to have resistance values (in ohms per metre) of :- - 17 -

Solution:- - 17 -

EVALUATION - 19 -

CONCLUSION - 19 -

BIBLIOGRAPHY - 20 -

Books - 20 -

Catalogues - 20 -

Websites - 20 -

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Task 1

1.1 The diameter in millimetres of a reel of wire is measured in 48 places and the results are as shown.

2.10 2.29 2.32 2.21 2.14 2.222.28 2.18 2.17 2.20 2.13 2.132.26 2.10 2.21 2.17 2.28 2.152.16 2.25 2.23 2.11 2.27 2.342.24 2.05 2.29 2.18 2.24 2.162.15 2.22 2.14 2.27 2.09 2.212.11 2.17 2.22 2.19 2.12 2.202.23 2.07 2.13 2.26 2.16 2.12

(a) Form a frequency distribution of diameters having 6 classes.

Solution:-

Range of Data = Highest Value – Lowest Value= 2.34 – 2.05= 0.29

Size of each Class =

=

= 0.048 approx

(Therefore increment each class by 0.04)

Frequency Distribution for the Diameters:

Class Class Mid-Point Frequency2.05 – 2.09 2.07 32.10 – 2.14 2.12 112.15 – 2.19 2.17 112.20 – 2.24 2.22 122.25 – 2.29 2.27 92.30 – 2.34 2.32 2

4


(b) Draw a histogram depicting the data.

Solution:-

5


(c) Form a cumulative frequency distribution.

Solution:-

Class Frequency Upper Class Boundary

Cumulative Frequency

2.05 – 2.09 3 Less than 2.095 32.10 – 2.14 11 Less than 2.145 142.15 – 2.19 11 Less than 2.195 252.20 – 2.24 12 Less than 2.245 372.25 – 2.29 9 Less than 2.295 462.30 – 2.34 2 Less than 2.345 48

(d) Draw an ogive for the data.

Solution:-

6


Task 2

2.1 Determine the mean, median and mode values for the following set of data correct to 1 decimal place:

73.8 126.4 40.7 141.7 28.5 237.4 157.9

Solution:-

Mean

Median

Mode = This set has no Mode!

7


2.2 The values of capacitance in microfarads selected at random from a large batch of similar capacitors:

34.3 25.0 30.4 34.6 29.6 28.7 33.4 32.7 29.0 31.3

Determine the standard deviation from the mean for these capacitors, correct to 3 significant figures.

Solution:-

Standard Deviation

25.0 30.9 5.9 34.8128.7 30.9 2.2 4.8429.0 30.9 1.9 3.6129.6 30.9 1.3 1.6930.4 30.9 0.5 0.2531.3 30.9 0.4 0.1632.7 30.9 1.8 3.2433.4 30.9 2.5 6.2534.3 30.9 3.4 11.5634.6 30.9 3.7 13.69

to 3 significant figures

8


Task 3

3.1 In an experiment to determine the relationship between the current flowing in an electrical and the applied voltage, the results obtained are:-

Current (mA) 5 11 15 19 24 28 33

Applied voltage (V) 2 4 6 8 10 12 14

Determine using the product moment formula the coefficient of correlation for these results.

Solution:-

Values of r should be between -1 and +1

Let X be the Variable Current Values.Let Y be the Variable Voltage Values.

5 2 19.3 14.3 8 611 4 19.3 8.3 8 415 6 19.3 4.3 8 219 8 19.3 0.3 8 024 10 19.3 4.7 8 228 12 19.3 8.7 8 433 14 19.3 13.7 8 6

85.8 204.49 3633.2 68.89 168.6 18.49 40 0.09 0

9.4 22.09 412.7 75.69 1682.2 187.69 36

9


Therefore:

Or

to 3 decimal places

Very good correlation exists between the values of Current (mA) and Applied Voltage (V).

10


3.2 The relationship between the voltage applied to an electrical circuit and the current flowing is as shown:

Current (mA) Applied voltage (V)

2 5 4 11 6 15 8 1910 2412 2814 33

Assuming a linear relationship, determine the equation of the regression line of applied voltage, Y, on current X, correct to 4 significant figures.

Solution:-

Let X be the Independent Current Values.Let Y be the Dependant Voltage Values.

Current (mA), X Voltage (V), Y X 2 XY Y 2 2 5 4 10 254 11 16 44 1216 15 36 90 2258 19 64 152 361

10 24 100 240 57612 28 144 336 78414 33 196 462 1089

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By substituting these values into the equations gives:

(1)(2)

Multiply (1) by 56 to give:

(3)

Multiply (2) by 7 to give:

(4)

Deduct (3) from (4), leaving:


Substitute a1 into (1) to give:


Therefore:

12


Check:-

13


Task 4

4.1 A box of fuses are all of the same shape and size and comprises 23 off 2A fuses, 47 off 5 A fuses and 69 off 13 A fuses. Determine the probability of selecting at random :-

(a) A 2 A fuse

(b) A 5 A fuse

(c) A 13 A fuse.

X 2 Amp 5 Amp 13 AmpY 23 47 69

Solution (a):-


Solution (b):-


Solution (c):-


Check:- that the sum of a, b and c is equal to 1:

14


4.2 An automatic machine produces on average 10 % of its components outside of the tolerance required. In a sample of 10 components from this machine determine the probability of having three components outside of the tolerance required by assuming a Binomial distribution.

Solution:-

The binomial expansion of is:

Therefore the binomial expansion of is:

From this we can see that:

The probability of having zero defective components:

The probability of having one defective component:

The probability of having two defective components:

The probability of having three defective components:

15


4.3 In problem 4.2 determine the probability of having three components outside of the required tolerance using the Poisson distribution.

Solution:-

As long as , a good approximation of the Binomial Distribution can be given using the Poisson Distribution.

Therefore the Poisson Distribution Expression can be used.

The probability of having zero defective components:

The probability of having one defective component:

The probability of having two defective components:

The probability of having three defective components:

Check:-

The Poisson Distribution is a method of approximation, therefore when comparing the answers to the Binomial Distribution (in answer 4.2) there is a slight dissimilarity!

16


4.4 The specific resistance of a reel of silver wire of nominal diameter 0.5 mm is estimated by determining the resistance of 7 samples of the wire. These were found to have resistance values (in ohms per metre) of :-

1.12 1.15 1.10 1.14 1.15 1.10 and 1.11

Determine the 99 % confidence interval for the true specific resistance of the reel of wire.

Solution:-

Firstly I need to work out the Standard Deviation of this set of numbers:

1.10 1.124285714 0.024285714 0.000589795904489796

1.10 1.124285714 0.024285714 0.000589795904489796

1.11 1.124285714 0.014285714 0.000204081624489796

1.12 1.124285714 0.004285714 0.000018367344489796

1.14 1.124285714 0.015714286 0.000246938784489796

1.15 1.124285714 0.025714286 0.000661224504489796

1.15 1.124285714 0.025714286 0.000661224504489796

17


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The Confidence Coefficient for 99% is: 2.58 (determined from the John Bird Book, Page 582)

Therefore:

i.e. The 99% Confidence Interval is between:

Ω/m

And

Ω/m

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Evaluation

During this statistics assignment I found a large amount of information in the Higher Engineering Mathematics Book, by John Bird.Question one with the Frequency Distribution and Histogram, had been covered during our work in the National Certificate last year. This allowed me to check over previous work to ensure I was using the correct methods to get the answer. I can usually handle statistical data well when in excel, and can manipulate graphs to look the way I want them to look, however I had difficulty in creating the Cumulative Frequency Curve. I found it difficult to make the axis state the correct numbers without it affecting the line of best fit.Question two was again a general question that I have done work on before in previous years, so solving the problem was possible by reviewing my old notes.Question three was worked through and solved by intensively following the Engineering Mathematics Book. The book had a very clear guide of gathering and calculating the required information to put into the formula. I then had to calculate the equation of the line, this was done using class notes.I found Question four interesting as it is the mathematical answer to luck. By learning how to work out the probability you can obviously increase your chance of successfully picking the correct thing by comparing the likeliness of it coming up. Although it is unlikely I will ever apply them to a real life solution, it was interesting to see how things like the chances of winning the national lottery are calculated. I used two methods, with the Binomial method being more accurate than the Poseidon method. Confidence intervals were something I haven’t used before so had to do a bit of background reading to be able to successfully apply it to the question.

Conclusion

I am pleased with this assignment, as it involved searching for the appropriate methods by myself.I found it difficult to check the majority of the answers as, without further explanation, there is only one method. This means that only a few of the answers from the assignment have been checked.I enjoyed working out the Binomial Distribution task, as I had never heard of it before this assignment. It involved understanding the rearranging of formulae, to be able to follow the methods used in the John Bird Book, so it was good practice. Having understood it I feel confident the answer is accurate and correct.

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Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

Books

Higher Engineering Mathematics (John Bird) ISBN: 0-7506-8152-7

Catalogues

N/A

Websites

N/A

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BTEC HNC - Analytical Methods - Statistics and Probability

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