Exercise Economics Chapter 3 (Some Mathematical and Statistical Concepts)- Class 11 for KPK textbook board.
Q.1) Write notes:
i) Continuous variables ii) Discontinuous variables iii) Independent and dependent variables iv) Function v) Degree of an equation vi) Exponent vii) Coefficients viii) Linear equation ix) Quadratic equation x) Increasing function xi) Decreasing function xii) Parameter
Answer: i) Continuous variables: A Continuous variable is the one that assumes all values within its range is called a continuous variable. In a Continuous variable, the value of the variable is never an exact point. It is always in the form of an interval, the interval may be very small. For example, the age and height of a person. ii) Discontinuous variables: A discontinuous variable is the one that does not assume all values in its range is called a discontinuous variable. It takes up some values and leaves the other. Thus there is a gap in its values. For example, the price of Rice per kg changes from Rs.20 to 30 and Rs.30 to Rs. 40 and then to 50. iii) Independent and dependent variables : A variable which can assume any value independently is called an independent variable. If the change in the value of an independent variable brings a change in the value of another variable, that will be the dependent variable. For example, the case of price and quantity supplied. According to the law of supply, quantity supplied increases with an increase in price and vice versa. Here, the price changes independently and quantity supplied changes due to change in price. Therefore, the price is an independent variable whereas the quantity supplied is the dependent variable. iv) Function: A function is defined as a relationship between two variables such that for on value of the first variable, there is only one value of the second variable. In other words, if there exists one to one (1 -1) correspondence between two variables, then the relationship between the variables would be – ‘Functional relationship”. e.g. y= f(x) In this equation, for one value of x, there exists a unique value of y. Therefore we read it as “y is a function of x”. v) Degree of an equation: The maximum or larger power in the equation is known as the degree of the equation. For example, ax2+bx+c=0 is equation of degree 2. vi) Exponent It is also known as index or power. It is a quantity that is written over the head of the variable e.g x4. It means that x is multiplied four times x.x.x.x= x4. vii) Coefficients: Co-efficient is the quantity that appears on the left side of the unknown. For example, 2x2+3x+5=0. 2 and 3 are co-efficient. viii) Linear equation If the maximum power of the variable is one then the equation is said to be linear. We have a standard linear equation as: ax+b=0 where ‘a’ and ‘b’ are constants. Examples are 4x+3=11. ix) Quadratic equation: If the maximum power of the variable is two then the equation is said to be quadratic. We have a standard quadratic equation is: ax2+bx+c=0, where ‘a’, ‘b’ and ‘c’ are constants and a≠0. Examples are x2+2x-9=0. x) Increasing function Whenever two variables have a functional relationship i.e y=f(x), when the value of one variable increases, the value of other variables also increases or when one variable has decreasing tendency, the other variable also has a decreasing tendency or both variables have same tendency to change shows an increasing function. For example y=2x+3. xi) Decreasing function: Whenever two variables are related in the sense that one variable has an increasing tendency, while the other variable has a decreasing tendency or there exists a negative correlation between the two, it will be a decreasing function. For example y=20-3x. xii) Parameter There are some variables whose values keep on changing in real life but they are assumed to remain constant for illustration of economic laws. They are called parameters.
Answer: A relation is a set of inputs and outputs that are related in some way. When each input in relation has exactly one output, the relation is said to be a function. To determine if a relation is a function, we make sure that no input has more than one output.
Q.3) Discuss methods of collecting data.
Answer: Methods of collecting data are as follows: i) Questionnaire Method ii) Direct personal observations iii) Indirect personal observations iv) Official Statistics v) Collection through enumerators
Q.4) Write notes on:
i) Tabulation ii) Classification of data.
Answer: i) Tabulation: The orderly arrangement of data into various rows and columns is called tabulation. Tabulation is a stage where data are ready for reading, for quick understanding, for publication, and for further statistical work. When the qualitative or quantitative raw data are classified according to one characteristic or one variable, the tabulation is called single or one-way. The tabulation is two-way when there are two variables. Similarly the tabulation is manifold when the data are divided into different categories on the basis of more than two criteria. ii) Classification of data: “Classification is the process of arranging things in groups or classes according to their resemblances and affinities”. or The process of arranging data in groups or classes according to resemblances and similarities is called classification. The classification of the data mainly depends upon the nature, scope, and purpose of the statistical inquiry. Some characteristics of a good classification are: (1) The classification should be unambiguous (2) The classification should be stable (3) The classification should not be rigid.
Q.5) Make a hypothetical table using two-way classification of data about Intermediate students of Lahore (Make the classes as science / Arts and Male / Female).
Q.6) Make the time series for years 2010-2015 for number of cars in Pakistan (Values in millions).
Q.7) Explain the concept of function. What are kinds of functions?
Answer: Function: A function is defined as a relationship between two variables such that for on value of first variable there exists one and only one value of the second variable. In other words, if there exists one to one (1 -1) correspondence between two variables, then the relationship between the variables would be – ‘Functional relationship”. e.g. y= f(x). Kinds of function are: i) Increasing function Whenever two variables have a functional relationship i.e y=f(x), when the value of one variable increases, the value of other variables also increases or when one variable has decreasing tendency, the other variable also has to decrease tendency or both variables have same tendency to change shows an increasing function. For example y=2x+3. ii) Decreasing function: Whenever two variables are related in the sense that one variable has an increasing tendency, while the other variable has a decreasing tendency or there exists a negative correlation between the two, it will be a decreasing function. For example y=20-3x. Single Valued Function: In the classical terminology, the function corresponds to a Valued Function ” where for one value of independent variable i.e. x In y = f(x) there will be one single value for the dependent variable i.e. Y. Therefore, this one to one correspondence shows the single-valued function. Multi-Valued Function : In the older (classical) terminolog} relation corresponds to .0.valued function in which one value of independent variable results in more than one value (multi-values) of the dependent variable. This is the multi-valued function. For example Y2 = X2. Implicit Function: Whenever two variables are dependent upon each other and by putting a value of one (X or Y) variable we get the value of the other variable i.e. (X or Y), this function is said to be implicit function. This term should be written in the form, F(x,y)=0 y-f(x)=0 For example xy=12 Explicit Function : If a function is defined in a sense in which one is the dependent variable and the other is the independent variable, it is called an explicit function. The funtion Y= f(x) and x=g(y) are called explicit functions. For example, Y = 3x + .2 is an explicit function as Y is depending upon X. Inverse Function: Whenever we interchange the dependent and independent variables of a function Y = f(x), we get inverse function. i e X = f-1(y). This will be read as X is an inverse function of Y. For example, xy=5 gives y =5x-1. Constant Function: A function whose range consists of only one element is called a constant function. We can write the function Y = f(x) = 9. Hence the value of Y remains the same as 9, regardless of the value of X. In economics, we use T F C and autonomous investment as examples of this function. Linear Function : A function Y = f(x) of a single variable is linear if it takes the form. Y = f(x) = ax + b. where a, b are constants e.g. Y = 4x + 9 Quadratic Function : A function Y=f(x) is called quadratic function if it takes the form. Y = f(x) = ax2+ bx + c e.g. = x2+ 6x + 5. Log Function: When a variable is expressed as a function of the logarithm of another variable, the function is referred to as logarithmic function. For example Y = log10x and Y= loge x(In x). Exponential Function: If a variable appears in the exponent or power of the fixed base. i.e. 2x, the function is called the exponential function. e.g. Y= 2x or Y = Cx.
Q.8) Express the following statements in functional notation and give specific forms if possible.
i) Profit is a function of the quantity of output (Q). Each unit gives Rs. 10/- as profit ii) Utility (U) is a function of quantity consumed (Q) iii) Saving (S) is a function of income (Y). People save 20 % of their incomes.
Answer: i) P=f(Q) = 10Q ii) U=f(Q) iii) S=f(Y), S=20% of Y S=(20/100) × Y S=(1/5)× Y
Q.9) Make the graphs of the equations 2x – 3y = 20, Q – 2 P = 20, Y = 10 + x2.
Q.13) Explain the method for constructing :
i) Simple price index number. ii) Weighted price index numbers
Answer: i) Simple price index number: A simple index number measures a relative change in single variable with respect to the base. A simple index number is the ratio of two values representing the same variable, measured in two different situations or in two different periods. For example, a simple index number of the price will give the relative variation of the price between the current period and a reference period. Weighted price index numbers: An index number that measures a relative change in a group of variables keeping in view the relative importance of variables. There are two methods of calculating weighted index numbers. i) Weighted aggregative price index numbers Price index= (Current year expenditure/ Base year expenditure) x 100 ii) A weighted average of relatives price index numbers: Pon= ∑WI/∑W where I=pn/po x 100 and W=poqo