MATH 130 – BASIC STATISTICS
Question 1
a) (i)As the phrases suggest the main difference is that one deals with “quantity” which is actually measurable while the other deals with “quality “/description of the data. Quantitative data is that represented in numbers and is thus measurable (The data collected for a numeric variable) this may involve data collected by direct/physical counting, interviews, questionnaires and other psychometric tests. Qualitative data on the other hand is that information obtained by describing meaning rather than statistical deductions (The data collected for a categorical variable) e.g. through case studies, interviews.
(ii)Continuous variables are value based (measurable) within a range. The values may be as tiny as the measurement tool allows e.g. age, time height, length etc. Discrete variables on the other hand take values based on counts from a set of different entire values i.e. they are wholesome it therefore cannot take the form of a fraction or decimal e.g. the population of men, women or children in an area, the number of cars, businesses all of which take whole units.
b) Statistics play a vital role in nearly all businesses and form the backbone for all future development strategies and control. It plays a pivotal role of determining whether the business is on the right track, if its achieving its targets and set goals and also in determining what needs immediate attention or remedial action if things are not working out well. They therefore serve as an eye opener to businesses thus helping them avert disastrous situations which are achieved by analyzing data and statistics. It will be noted that every business plan starts with an extensive research which is compiled into statistics which often influence the final decision. Such statistics are very useful in situation of acquiring finances from banks, investors and other financial institutions; they also aid in development and learning; staff requirement employment; and therefore it is not easy to work without detailed statistical reporting as the businesses would not keep track of finances and would lose direction. Statistics provide business people with the tools they need to implement short and long-term processes, plans, and marketing initiatives. Profit and loss statements, customer management information, and market share graphs are just a few business tools derived from the use of business statistics. Without these the managers may not have full knowledge of essential information necessary for them to be successful.
c) (i) Properties of a good average
• Averages should be rigidly defined. If it is so, instability in its value will be no more and would always be a definite figure. If an average is left to the estimation of an observer and if it is not a definite and fixed value it cannot be representative of a series. The bias of the investigator in such cases would considerably affect the value of the average.
• An average is said to be a true preventative only when it is based on all the values of a variable otherwise, it cannot considered a good average. It should be based on all the observations of the series. If some of the items of the series are not taken into account in its Calculation the average cannot be said to be a representative one.
• It should be capable of further algebraic treatment. If it not then its use is bound to be very limited. It will not be possible to calculate e.g. the combined average of two or more series from their individual averages or even to study the average relationship of various parts of a variable if it is expressed as the sum of two or more variables.
• It should be easy to calculate and simple to follow. If the calculation of the average involves tedious mathematical processes it will not be readily understood and its use will be confined only to a limited number of persons. It can never be a popular average.
• It should not be too abstract or mathematical and there should be no difficulty in its calculation. They should be such that they can be easily understood by persons of ordinary intelligence.
• It should not be affected by fluctuations of sampling. If two independent sample studies are made in any particular field which are homogenous, the averages obtained, should not materially differ from each other e.g. if different samples are taken from the production of rice, the mean of these samples should be equal.
• A good average is one which is not affected by biasness in the distribution. Contrary to this, if it is affected by biasness, it cannot become a true representative.
(ii)The average should be representative of the distribution which enables the person using it to comprehend in a single effort. Some other averages which do not take into account this property and hence all the values of a group are not satisfactory averages.
Question Two
(i)
= 6 + 28 + 36 + 20 + 13
5
= 20 People
(ii) √ (20-5)² + (20-28)² + (20-36)² + (20-20)² + (20-13)²/5
√113 = 10 people
(iii) Mode = 20 – 30 Consumption (in K-watt hours)
(iv) Coefficient of skewness = Arithmetic mean – Mode/ Standard deviation
= 20-36/10
= -1.6
Question Three
a) A correlation is a single number that describes the degree of relationship between two variables. It is therefore a statistical measure of how two variables move in relation to each other.
Positive Correlation is where the variables moves in the same direction if one variable increases the other one also increase and vice versa. Negative Correlation on the other hand is the correlation in opposite direction, in that when one variable increases the other decreases and vice versa, if there exists no relationship between the two variables such that the value of one variable change and the other variable remain constant then that is a zero correlation.
b) The formula for the correlation is:
(i) R = N∑XY –(∑X )(∑Y )/ √[N∑X² -(∑ X)²][N∑ Y² -(∑ Y)²]
Where: R is the correlation coeficient
N is the number of pairs = 10
X is the Aptitude Score and
Y is the productivity index
X Y XY XY² X² Y²
60 68 4,080 16,646,400 3,600 4,624
62 60 3,720 13,838,400 3,844 3,600
65 62 4,030 16,240,900 4,225 3,844
70 80 5,600 31,360,000 4,900 6,400
72 85 6,120 37,454,400 5,184 7,225
48 40 1,920 3,686,400 2,304 1,600
53 52 2,756 7,595,536 2,809 2,704
73 62 4,526 20,484,676 5,329 3,844
65 60 3,900 15,210,000 4,225 3,600
82 81 6,642 44,116,164 6,724 6,561
650 650 43,294 206,632,876 43,144 44,002
∑X ∑Y ∑XY ∑XY² ∑X² ∑Y²
R = 10(43,294) –(650 )(650 )/√[10(4,3144) -(650)²][10(44,002) -(650)²]
R= 0.834
(ii) Regression equation is Y’ = a + bx
To find a and b using the above table:
Therefore a = (650×43,144) – (650 x 43,294)/ 10(43,144)- (650×650) = -10.91
b= 10(43,294) – (650 x650 )/ 10(43,144) – (650 x650) = 1.17
Y’ = -10.91+1.17X
(iii) Aptitude Score (X) = 92
Then Y = -10.91 + (1.17 x 92)
Productivity Index (Y) = 96.7
