Mathematics Methods and Statistical Techniques

Part 1 

The power (P) dissipated in a resistor (R) which is subjected to a voltage (V) across its terminals is given by the well-known formula; 

Since you are new to the company your new Line Manager is unsure of your capabilities, and has asked you to use the dimensions of P and V to determine the dimensions of R 

Solution: 

Make R the subject of the formula: 

A guitar string, made by your Musical Instruments division, has mass (m), length ( 𝑙𝑙 ) and tension (F). It is proposed by one of your junior colleagues that a formula for the period of vibration (t) of the string might be; 

Use dimensional analysis to show your colleague that this formula is incorrect. 

Solution: 

For this formula to be correct: 

Ignore since it is just a number (Rule 1). 

We can use the “proportionality” (α) symbol instead of an equals sign to represent the proposition. 

This proposed formula has dimensions of , which are clearly not the dimensions of time ( period of vibration (t) of the string). Time has a dimension of T. Therefore, the formula is incorrect. 

An analogue-to-digital converter (ADC), manufactured by your Signals division, takes 20 voltage samples of a ramp waveform, measured in mV, as follows… 2, 4, 6, 8, 10 …40 

Your Test colleague has asked you to assist by using a formula to calculate the sum of these 20 voltage samples. 

Solution: 

This is an arithmetic progression: 

The common difference is 2 

The formula used to calculate the sum of n terms in an arithmetic progression is: 

Where, 

Check: 

A digital chip, made by your Microelectronics division, counts continuously in the sequence: 1024, 2048, 4096, 8192, … 

You have been asked to use a formula to calculate the 9th count of the chip. 

Solution: 

This is a geometric series: 

The formula used to calculate term in a geometric series is: 

Where, 

Check: 

A series electrical circuit which you are testing features a capacitor (C) charging via a 1MΩ resistor (R) and a 12V dc supply (Vs). The voltage across the capacitor (Vc) may be described by the equation… 

Where t represent time. 

Assuming that Vc is 2V after a time of 4 seconds, determine the approximate value of the capacitor. 

Solution: 

But, 

One of your commonly-used laboratory instantaneous test signal voltages (vs) is described by the equation… 

Where f=1MHZ and t represents time. 

Make time (t) the subject of this formula, and hence determine the first point in time when the instantaneous signal voltage has a magnitude of +3V. 

Note: A colleague has reminded you that you need to have your calculator in radians mode (RAD) for this calculation, because the angle is given in radians (i.e. π is featured). 

Use suitable software to draw at least two cycles of this signal, and annotate the drawing so that non-technical colleagues may understand it. 

Solution: 

Note: 

Therefore; 

Hence, 

Therefore, the first point in time when the instantaneous signal voltage has a magnitude of +3 is 0.2083 s. Using Desmos online graphic tool (available at https://www.desmos.com/ ) , the cycles for this signal was drawn which is as shown below: 

The curve assumed by a heavy power cable, manufactured by your Power division, is described by the equation… 

Where x and y are horizontal and vertical positions respectively. Calculate; 

The value of y when x is 104. 

The value of x when y is 180. 

Solution: 

In this case, 

Therefore, 

The value of y when x is 104. 

Here, 

The value of x when y is 180. 

Here, 

You are testing a decorative clock, to possibly be manufactured by your Consumer Electronics division, and attach a mass (m) to a string of length ( 𝑙𝑙 ) to form a simple pendulum. Assuming that the acceleration due to gravity (g) of the earth may have an influence on the period (t) of the pendulum swing, use dimensional analysis to find a formula for t which could possibly involve m, 𝑙𝑙 , and g. 

Solution: 

Variable	Units	Dimensions
m	kg	M 1 L 0 T 0
l	m	M 0 L 1 T 0
g	m/s^2	M 0 L 1 T -2
t	s	M 0 L 0 T 1

The assumption made is that g has an influence on t. t is a function of mass, length and gravity. 

Therefore, 

Solving these two equations gives use the values of b and c. 

Π term becomes. 

Make t the subject of the formula. 

You are testing a prototype loudspeaker for your Audio division, which may possibly be used in smoke-filled environments when nuclear reactor emergencies occur. A senior colleague has asked you to use dimensional analysis to predict how the speed of sound (u) in a gas may be influenced by the gas pressure (p), gas density (r) and acceleration due to gravity (g). 

Solution: 

Let the speed (u) of light be dependent on the following factors: 

Therefore, 

Variable	Units	Dimensions
u	m/s	M 0 L 1 T -1
p	N/m^2	M 1 L -1 T -2
r	Kg/m^3	ML -3 T 0
g	m/s^2	M 0 L 1 T -2

K is a constant. Its dimensional values are: 

Therefore, the equation becomes: 

Comparing the powers of M, L, and T 

Solving (I) and (II) simultaneously gives us the values of a and b. 

Substituting the value of a in (III) gives us the value of c: 

Therefore; 

Part 2 

Your Communications division manufactures wireless dongles for use in general computing. These dongles have a maximum allowed radiative power of +20dBm. A random sample of ten dongles was taken, and their transmit power was measured by a colleague, using a spectrum analyser. The results are as follows; 

Sample	1	2	3	4	5	6	7	8	9	10
Power (+dBm)	18.1	19.2	18.4	18.1	19.9	18.1	17.4	19.1	18.1	17.4

Calculate the mean transmit power for these samples. 

Determine the standard deviation for the samples. 

Sample	Power (+dBm)	Power-X	(Power-X)^2
1	18.1	-0.28	0.0784
2	19.2	0.82	0.6724
3	18.4	0.02	0.0004
4	18.1	-0.28	0.0784
5	19.9	1.52	2.3104
6	18.1	-0.28	0.0784
7	17.4	-0.98	0.9604
8	19.1	0.72	0.5184
9	18.1	-0.28	0.0784
10	17.4	-0.98	0.9604
Count (n)	10
Mean (X)	18.38	Total	5.736

Alternatively, the mean and standard deviation for the samples can be determined using excel functions. 

Sample	Power (+dBm)
1	18.1
2	19.2
3	18.4
4	18.1
5	19.9
6	18.1
7	17.4
8	19.1
9	18.1
10	17.4
Total	183.8
Mean	18.38
Standard Deviation	0.7574

Produce a Tally Chart showing the frequency of measured transit powers. 

Power (+dBm)	Frequency
17.4	2
18.1	4
19.1	1
19.2	1
19.9	2

You visit your Manufacturing division, which has a machine producing metal bolts. In a tray of these bolts, 94% are within the allowable diameter tolerance value. The remainder exceed the tolerance. You withdraw six bolts at random from the tray. Determine the probabilities that; 

Two of the six bolts exceed the diameter. 

More than two of the six bolts exceed the diameter. 

Solution: 

Two of the six bolts exceed the diameter. 

The part of the binomial in bold, above, will give us the probability that two of the six bolts, (p (2)), exceed the diameter. 

More than two of the six bolts exceed the diameter. 

The part of the binomial in bold, above, will give us the probability that more than two of the six bolts, (P (>2)), will not exceed the diameter. 

Your Components division manufactures capacitors, and the mean capacitance of 400 capacitors you have selected is 100µF, with a standard deviation of 7µF. If the capacitances are normally distributed, determine the number of capacitors likely to have values between 90µF and 110µF. 

Note: Use the z-table given in Appendix A when answering part c. 

Calculate z values; 

From the z-table, this corresponds to an area of 0.0557 of the total area under the function (which is 1). 

From the z-table, this corresponds to an area of 0.0557 of the total area under the function (which is 1). 

The total area between the limits 99 and 101 is therefore twice 0.0557. 

Since we have 400 capacitors in the sample, then the number of capacitors we expect to have capacitance in the range 99 to 101 is: 

A colleague, who is a Fuels engineer, is testing the effects of an experimental fuel additive for petrol engines which your company is developing. She adds the same sample amount of additive to 100 full petrol tanks for the same model of car, and records the number of miles per gallon (mpg) for each car after being driven around a test track at a constant speed, until the fuel runs out. She knows that such testing undertaken without the additive produces a mean mpg figure of 44. Collecting results with the additive, she notices that the mean mpg figure is 48 with a sample standard deviation of 13 mpg. 

By interpreting the results of the testing, show whether you agree, or not, with her hypothesis that the fuel additive has influenced the number of miles per gallon for the cars. 

Draw by hand, or use suitable software, to produce a graphic, suitable for a nontechnical company executive, which represents the results of your analysis 

Note: Use the z-table given in Appendix A when answering part d. 

Solution: 

There are two hypotheses: 

Let us assume that is true. 

We know that; 

So, z is 3.1 standard deviations away from the normalized center. Using this z-table, this value corresponds tan area of 0.49903. This figure only represents the area to the right of the center of the Normal curve. So, we must find the total are under the curve, which is: 

Since the total area under the standardized normal curve is 1, the area occupied by our z-value is: 

The chances the null hypothesis is true is: 

Therefore, we reject the null hypothesis and accept the alternative hypothesis.