Question

OUESTİON ONE (a) Explain the following terms: ) An outcome ) Hypothesis O Dummy Activity I mark) 1 mark I mark [I mar'k I mr dvButfer stock A network Coetciut of variation on's coefficient of skewness for variation is 40%. Its mode is 80, Find distribution is 0.5, and the coefficient of the mean and median of the distribution 13 marks (c) The lifetime of car batteries produced by a company are normally distributed with mean 16,000 hours and standard deviation of 4000 hours. A battery is selected at random from 4000 batteries. Determine the number of batteries whose lifetime would be c) The lifetime of car batteries produced by a company are normally distributesd ) Less than 12,000 hours (ii) Between 14,000 and 21,000 hours ii) More than 23,000 hours [2 marks] [2 marks)] [2 marks 3 marks (iv) Between 19,500 and 11,9000 hours

Answer 1

(a)

(i)

An outcome is a possible result of an experiment. For example, Rolling of a die has 6 possible outcomes and flipping a coin has two possible outcomes. Each possible outcome of a particular experiment is unique and different outcomes are mutually exclusive, that is, only one outcome will occur on each trial of the experiment.

(ii)

A hypothesis is an assumption about a population parameter such as population mean, population proportion or population variance, etc,. or a statement about two or more population parameters being equal. The assumption may or may not be true and so, hypothesis is tested to conclude if it is true.

For example, the statement that a population proportion is 0.30 is a hypothesis and researcher conducts a statistical study to test the validity of this hypothesis.

(iii)

A Dummy activity is an imaginary activity. It does not actually exist in the project activities and often have a zero completion time. They are used to represent precedence relationships that cannot be easily, if at all, represented using the actual activities of the project. By convention, dummies are always indicated by the dotted arcs in the network diagrams.

(iv)

Buffer stock, also known as safety stock, is usually a stock of any commodity that is purchased and stored, usually by a Government at periods of excess suppy to prevent prices falling below a certain level and released during shortage of such commodity to prevent prices rising above a certain level. It's a strategy used by Governments to stabilize prices in the market (generally, in the commodity market).

(v)

A network is a set of different elements or components connected together for the purpose of sharing resources and data by communicating with each other. For example, A computer network is a set of computers connected together.

(vi)

Coefficient of variation, CV is the ratio of the standard deviation to the mean. It shows the extent of variability in relation to the mean. It is expressed as a percentage.

CV =(standard deviation/mean)*100

(b)

Coefficient of variation, CV =40% =0.40

CV =$\sigma/\bar{X}$ =0.40 $\implies$$\sigma =0.40 \bar{X}$

Pearson's Coefficient of skewness=

(mean - mode)/std.deviation =$(\bar{X} - Mo)/ \sigma$ =0.5

$\implies$$(\bar{X} - 80)/\sigma$ =0.5​​​​ $\implies$$\bar{X} - 80= 0.5 \sigma$

Substituting, $\sigma =0.40 \bar{X}$ , we have:

$\bar{X} - 80 =0.5(0.40 \bar{X})$$\implies \bar{X} - 80 =0.20\bar{X}$$\implies \bar{X}(1-0.20)= 80 \implies 0.80 \bar{X} =80 \implies\bar{X} =80/0.80 =100$​​​​​​

Mean =100 and mode =80

Emperical rule is: Mode =3 Median - 2 Mean, that is, 3 Median =Mode+2 Mean

So, Median =1/3(mode+2 mean) =1/3[80+2(100)] =93.33

(c)

Given:

$\mu =16000\ hours \ and \ \sigma =4000\ hours$

N =4000 batteries.

(i) X<12,000 hours:

Z =$(X - \mu)/\sigma$ =(12000 - 16000)/4000 = - 1

P(X<12000) =P(Z< - 1) =0.5 - 0.3413 =0.1587

Number of batteries whose life time is less than 12,000 hours =4,000*0.1587 =635

(ii) 14,000 < X < 21,000 hours:

Z1 =(14000 - 16000)/4000 = - 0.5

Z2 =(21000 - 16000)/4000 = 1.25

P(Z1 > - 0.5) =0.1915 and P(Z2 < 1.25) =0.3944

P(14000<X<21000) =P(X>14000)+P(X<21000)

=P(Z1>-0.5)+P(Z2<1.25) =0.1915+0.3944 =0.5859​​​​​​

Number of batteries =4000*0.5859 =2344​​​​​​