SUBJECT :- STATISTICS Standard :- 12th Commerce

Answer the following question in one sentence

1) Which index number is used in India to find the rate of dearness allowance?

Ans: - Cost of living index number is used in India to find the rate of dearness allowance.

2) What is the main limitation of Scatter diagram method?

Ans: - The main limitation of scatter diagram method is that we cannot obtain the exact

measure of strength of correlation.

3) Define Regression coefficient.

Ans: - The co-efficient of 𝑏 𝑜𝑓 𝑥 in the linear regression model 𝑦 = 𝑎 + 𝑏𝑥 is known

as co-efficient of regression. OR The approximate change in the value of dependent

variable for a unit change in the value of independent variable. h represents the

appropriate increase or decrease in the value of Y per one unit increase or decrease in

the value of X.

4) State the interval required to know the trend in agricultural yield or industrial production.

Ans: - To require the trend in agricultural yield or industrial production. 10-15 years time

interval is required.

5) State the formula for the probability of occurrence of at least one event out of three events

A, B and C.

Ans: - Occurrence at least one event out of three evens A, B and C or 𝐴 𝑈 𝐵 𝑈 𝐶.

𝑃(𝐴 ∪ 𝐵 ∪ 𝐶)

= 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) — 𝑃(𝐴 ∩ 𝐵) — 𝑃(∩ 𝑐) — 𝑝(𝐵 ∩ 𝑐) + 𝑃(𝐴

∩ 𝐵 ∩ 𝑐)

6) Define discrete random variable.

Ans: - A random variable X, which is capable of taking any real value of set R or its

subset is called a discrete random variable

7) What is the probability that a continuous random variable takes definite value ?

Ans: -. The probability that a continuous random variable takes definite value is always

'0; (zero)

8) For a normal distribution, the estimated value of quartile deviation is 12. Find the value of

its standard deviation.

Ans: - In normal distribution fourth quartile =2

3𝜎,therefore 12 =

2

3𝜎 ∴ 𝜎 =

12×3

2= 18

Thus, the value of standard deviation is 18.

9) Explain the meaning of x.

Ans: -. If by decreasing the positive value of a variable X or by increasing negative value

of the variable X ; the value of x is brought very close to `0’. then it can he said that X

tends to 0. It is denoted by 𝑥 → 0. Here, 𝑥 → 0 means the value of 𝑥 approaches very

close to 0 but it will not be equal to 0

10) What is marginal revenue?

Ans: - The change in revenue due to small change in demand is called marginal revenue.

11) What is a base year ?

Ans: - When the price of an item is compared with the price of the same item in some

specific (fixed) year of the past, then that specific year is called the base year

12) Define correlation.

Ans: - If there are simultaneous changes in the values of two variables due to direct or

indirect cause effect then it is said that there is correlation between two variables

13) The regression co-efficient is independent of which transformation `?

Ans: - The regression co-efficient is independent from the change of origin.

14) Write the additive model of the time series

Ans: - The additive model of the time series is as follows:

𝑦𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝐶𝑡 + 𝑅𝑡, where yt = variable quantity, Tt = Trend, St =

Seasonal Component, Ct = Cyclical Component, Rt = Randon Component.

15) State the number of sample points of the random experiment of tossing six balanced

coins.

Ans: - The number of sample points in the random experiment of tossing six balanced

coins is𝑛 = 26 = 64.

16) State the relation between mean and variance of binomial distribution.

Ans: - In binomial distribution value of mean is always greater than its variance. Hence,

𝑛𝑝 > 𝑛𝑝𝑞 𝑂𝑅 𝑛𝑝𝑞 < 𝑛𝑝

17) Standard score is independent of unit of measurement". Is this statement true or false ?

Ans: - “Standard score is independent of unit of measurement." It is a true statement.

18) Mean of a normal distribution is 13.25 and its standard deviation is 10 Estimate the

value of its third quartile?

Ans: -" In a normal distribution 𝑄3 = 𝜇 + 0.675𝜎.

Therefore, 𝑄3 = 13.25 + 0.675(10) = 13.25 + 6.75 = 20

Thus, estimate value of its third quartile is 20.

19) Find the value of lim𝑥→3

𝑥4−81

𝑥−3

Ans: -lim𝑥→3

𝑥4−81

𝑥−3

lim𝑥→3

𝑥4 − 81

𝑥 − 3 [lim

𝑥→𝑎

𝑥𝑛 − 𝑎𝑛

𝑥 − 𝑎= 𝑛𝑛−1]

lim𝑥→3

= 4(3)4−1

lim𝑥→3

= 4(3)3

= 108

20) Find 𝑓(𝑥) for the function 𝑓(𝑥) = 21

Ans: - Derivate of constant number is always zero. So, if function 𝑓(𝑥) = 21 then

𝑓′(𝑥) > 0

21) What is the main purpose of index number?

Ans:- The main purpose of index number is to measure percentage relative change in

value of a variable in current period in comparison with its value in base period.

22) Define Perfect Positive Linear Correlation Coefficient.

Ans:- If the values of two random variable vary in the same direction and if one unit

increase or decrease in one variable results in the increase decrease in some fixed

proportion in the other variable then the correlation between two variables is said to be

perfect positive correlation.

23) State the linear regression model that represents the relation between two random

variables Y and X.

Ans :- Linear regression model 𝑦 = 𝑎 + 𝛽𝑥 + 𝑢 where, 𝑎 and 𝛽 are constants and 𝜇 is

an error variable.

24) What is meant by long term component?

Ans:- The slow and steady influence on the values of a time series spread over a long

period of time is called long term component or trend.

25) If 𝐴 = { 𝑥 | 0 < 𝑥 < 1} 𝑎𝑛𝑑 𝐵 = {𝑥 |1

4≤ 𝑥 ≤ 3} 𝑓𝑖𝑛𝑑 𝐴 ∩ 𝐵

Ans:-

26) State the probability distribution for Bernoulli trials.

Ans:- The probability distribution for Bernoulli trials is Binomial probability distribution

27) What is normal curve?

Ans:- The probability curve of bell—shape is called the normal curve.

28) Why is the total area of the region bounded by the normal curve and X— axis taken as

1?

Ans:- The normal curve is a probability curve and total probability is 1. Hence, the total area

of the region bounded by the normal curve and X—axis is taken as 1.

29) What is the limit of the product of a function with a constant?

Ans:- The limit of the product of a function with constant is equal to the product of the

limit of the function with the same constant. That means lim𝑥→𝛼

𝐾, ∫ (𝑥) = 𝐾. 𝐿 where K is

the constant.

30) How the marginal cost can be obtained?

Ans:- marginal cost can be obtained by taking the derivative of the cost function with

respect to x. thus, when the product is x then margin cost = 𝑑𝑐

𝑑𝑥

31) Fisher's Index Number is called an ideal index number. Write two properties in support

of the statement.

Ans:- . (I) In this Index Number, the price and quantity of base year and current year are

considered. (2) This Index Number is Geometric Mean of Laspeyre's and Paasche's index

number.

32) Write the Spearman's formula to find r.

Ans :- The Spearman’s formula to find r is 𝑟 = 1 − 6𝛴𝑑2

𝑛(𝑛2−1)

33) State the methods for obtaining regression lines.

Ans:- There are two methods of obtaining regression lines.

(I) Scatter diagram method.

(2) Least squares method.

34) What is forecasting?

Ans:- Forecasting is the inference based on the study of observations made on the post

phenomena for some specified point of time of the future

35) Arrange P(A u B),P(A),P(A n B),O, P(A)+(B) in the ascending order.

Ans:- Arranging in the ascending order : 0, P (A n B), P(A), P (A v B), P(A) + P(B).

36) State the conditions that p(x) becomes a probability distribution.

Ans:- If for random variable X. (I) for all the values of X of variable 𝑋 𝑝(𝑥) ≤ 0 and

(2) 𝛴𝑝(𝑥) = 1 then p (x) is the probability distribution of variable X.

37) State the relationship among the quartiles in normal distribution.

Ans:- In the normal Distribution two quartiles Q1 and Q3 are equidistant from the second

quartile Q2 = M. that means Q3 – M = M - Q1 -> Q3 + Q1=2M.

38) Mean of a normal distribution is 13.25 and its standard deviation is 10. Estimate the

value of its third quartile.

Ans:- In the normal distribution 𝑄3 = 𝜇 + 0.6745𝜎. therefore, Q3 = 13.25 +0.6745(10) =

13.25 +6.75 = 20. Thus , estimate value of its third quartile is 20.

39) If the value of the function is interminate then how can be we obtained the approximate

value of the function?

Ans:- if the value of the function is indeterminate limit can be used to approximate the

value of a function.

40) Give the meaning of increasing function.

Ans:- If h is a small positive number and if ∫ (𝑎) > ∫ (𝑎 + ℎ) and also ∫ (𝑎) > ∫ (𝑎 −

ℎ) then ∫ (𝑥) is said to be maximum at x = a.

41) State the formula of converting chain base into number into fixed base index number.

Ans:- 𝐹𝑖𝑥𝑒𝑑 𝑏𝑎𝑠𝑒 𝑖𝑛𝑑𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟 =(𝑐ℎ𝑎𝑖𝑛 𝑏𝑎𝑠𝑒 𝑖𝑛𝑑𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑒𝑎𝑟) × ( 𝑓𝑖𝑥𝑒𝑑 𝑏𝑎𝑠𝑒 𝑖𝑛𝑑𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑦𝑒𝑎𝑟)

100

42) State methods using in the student of correlation.

Ans:- (1) Scatter diagram method.

(2) Karl Pearson’s product movement method.

(3) Spearman’s rank correlation method.

43) How do you estimate dependent variable Y for the value of independent variable X?

Ans:- The estimate of demand variable Y for the value of independent variable X can be

done by regression line Y on X.

44) Give two examples of seasonal fluctuations.

Ans:- Two examples at seasonal fluctuations are

(1) Demand of woolens increases in winter.

(2) Increase in the sale of ready-made garments and fool-wears during the religious

Festivals.

45) Interpret the event A u B and A n B.

Ans:- . Event A u B means the occurrence of at least one of the events A and B. Event A n B

means the occurrence of event A and B both together.

46) By which condition parameter P of binomial distribution is negative skewness?

Ans:- In binomial distribution if 𝑝 >1

2, then there is negative skewness in the distribution.

47) When it is said that "Probability distribution is normal probability distribution"?

Ans:- If the curve of the probability density function of a random variable X is bell shaped.

then the probability distribution X is called the normal distribution

48) What is the value of mean and variance in standard normal distribution?

Ans:- The standard normal distribution mean = 0 and variance = 1.

49) How is the modulus of real number?

Ans:- Modulus of any real numbers is always non-negative.

50) What is the second order derivative?

Ans:- The second order derivative of the function means the derivative of the first order

derivative of the function. It is denoted by 𝑑2𝑦

𝑑𝑥2 𝑜𝑟 ∫ "(𝑥).

51) Why is Fisher's index number called an ideal index number"

Ans:- Following are the reasons for considering Fishers index number as an Ideal index

number :

(i) Geometric mean is used in its calculation.

(ii) It takes into consideration the price and quantity of both the base year and the

current year.

(iii) It satisfies the two important tests of index numbers.

(iv) It is free from bias.

52) Define 'Correlation Co-efficient'.

Ans:- The numerical measure of correlation between two variables is known as correlation

coefficient and it is denoted by r.

53) Why best In regression line method is known as least square method?

Ans:- In best lit regression line the line is selected for which the sum of square of Error's

approximation is minimum, so it is known as least square method.

54) In time series what is called long term fluctuations?

Ans:- The fluctuations which prevail on the time series for a long period of time is called

long term fluctuations or trend.

55) What is random event?

Ans:- An event depending on an unknown element (chance) is called a random event.

56) Write probability density function of binomial distribution.

Ans:- 𝑝(𝑋 = 𝑥) = 𝑃(𝑥) =𝑛 𝐶𝑥𝑃𝑥𝑞𝑛−𝑥 ,

Where, 𝑝 ≤ 𝑥 ≤ 𝑛, 𝑎𝑛𝑑 𝑝 < 𝑝 < 1

𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑠𝑓𝑢𝑙 𝑡𝑟𝑖𝑎𝑙𝑠.

𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑠𝑓𝑢𝑙 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝑡𝑟𝑖𝑎𝑙𝑠.

𝑝 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑞 = 1 − 𝑝.

57) Draw the graph of the normal curve and locate the positions of quartiles.

Ans:- Normal Probability Curve:

58) Which value of normal variable divides the area of normal curve in two equal parts?

Ans:- . Normal curve is symmetric on both the sides for mean of normal variable X. Therefore

for 𝑋 = 𝜇 of normal variable divides the area of normal curve into two equal pans.

59) When it should be said that limit of a function does not exist?

Ans:- When the value of x is brought nearer to a by increasing or decreasing its value, the

value of f(x) does not approach to a particular value then it should he said that bran of a

Function does not exist.

60) What is differentiation?

Ans:- The process of obtaining derivative of a function is called differentiation.

61) Explain the main difference between explicit weight and implicit weight.

Ans:- The weight that can be expressed in number is called explicit weight and the weight

that cannot be expressed in number is called implicit weight. In implicit weighing system

more varieties of commodities are taken to which more importance are to be given.

62) Interpret r = -I.

Ans:- When r = -1 the correlation between the variables is perfect negative. Here. an

increase in the value of one variable is associated with a proportional decrease in the value

of another variable.

63) What do you mean by error of approximation?

Ans:- The difference between given value of y and estimated value of y is called error of

approximation i.e. 𝑒 = 𝑦 − �̂�.

64) What is random variations?

Ans:- The values in the series change due to sudden and unpredictable causes, these

changes are called as random variations.

65) Is it possible that P(A) = 0.7 and P(A U B) = 0.45 for two events A and B in a sample

space?

Ans:- For two events A and B in a sample space P(A) = 0.7 and P (A u B) 0.45; it is not

possible because it is P(A) ≤ P(A u B) but here P (A u B) < P(A).

66) State the formula to find variance of discrete variable.

Ans:- Formula to find variable of discrete variable is as follows:

Variance:- 𝜎2 = 𝑉(𝑋) = 𝐸(𝑋2) − (𝐸(𝑋))2

Where, 𝛴(𝑋2) = ∑𝑥𝑖2 𝑝(𝑥𝑖) 𝑶𝑹 𝛴(𝑋2) = 𝛴𝑥2𝑝(𝑥)

67) Define standard normal variate.

Ans:- If a random variable X is a normal variate with mean is and standard deviation 𝜎.

then random variable 𝑍 = 𝑥 − 𝜇

𝜎 is called the standard normal variate.

68) For a probability distribution of standard normal variable, state the estimated limits for

the middle 50% observations.

Ans:- . For a probability distribution of standard normal variable, first quartile (Q1) = -0.675

and third quartile (Q3) = 0.675. Therefore, estimated limits for the middle 50% observation

is between - 0.675 and 0.675.

69) Define the real line?

Ans:- . The real line or real number line is a line where its points are the real numbers.

70) Find f(x) for the function f(x) = 50.

Ans:- Derivative of a constant number is always zero. So, if function f(x)=50 then f(x)=0.

71) Write the formula for obtaining the real wage?

Ans:- 𝑅𝑒𝑎𝑙 𝑤𝑎𝑔𝑒𝑠 = 𝑤𝑎𝑔𝑒

𝑐𝑜𝑠𝑡 𝑜𝑓 𝑙𝑖𝑣𝑖𝑛𝑔 𝑖𝑛𝑑𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟 × 100

72) State the names of different method of finding correlation.

Ans:- The different methods of finding correlation are : (1) Karl Pearson’s product moments

methods, and (2) Spearman’s rank correlation method.

73) What is the interval of co-efficient of determination?

Ans:- The interval of co – efficient of determination is between 0 and 1. 0 ≤ R2 ≤ 1.

74) Write the additive model of the time series.

Ans:- The additive model of the time series is as follows:

𝑦𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝐶𝑡 + 𝑅𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑦𝑡 = 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦, 𝑇𝑡 = 𝑇𝑟𝑒𝑛𝑑𝑠, 𝑆𝑡 =

𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡, 𝐶𝑡 = 𝐶𝑦𝑠𝑙𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡, 𝑅𝑡 = 𝑅𝑎𝑛𝑑𝑜𝑚 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡.

75) Write definition of elementary events.

Ans:- Events consisting of only a signal elements of a sample space U are called elementary

events. Elementary events are mutually exclusive and exhaustive events.

76) What is success and failure?

Ans:- If a certain result is obtained in a random experiment it is called success and is that

certain result is not obtained, it is called failure.

77) In the standard normal distribution write the estimated values of Q1 and Q3.

Ans:- In the standard normal distribution,

𝑄1 = 𝜇 − 0675 𝜎 𝑎𝑛𝑑 𝑄3 = 𝜇 − 0675.

78) How is the probability curve in standard normal distribution?

Ans:- the probability curve is bell – shaped in standard normal distribution.

79) Define neighborhood of a

Ans:- Any open interval containing, 𝑎, 𝑎 ∈ 𝑅 is called a neighborhood of a.

80) If y is a function x then how its differentiation can be shown?

Ans:- If y is a function of x, then is derivative is denoted by 𝑑𝑦

𝑑𝑥.

81) What is a base year?

Ans:- An index number measure the relative change in the value of a variable as compared

to some other period. The period with which the comparison is made is known as base

year.

82) Which is the original formula for calculating correlation coefficient given by Karl

Pearson?

Ans:- 𝑟 = 𝑐𝑜𝑣(𝑥,𝑦)

𝑆𝑥 × 𝑆𝑦

83) What is regression?

Ans:- Regression is the relation between two variable determined by an appropriate

mathematical function.

84) What are the components of short-term component?

Ans:- The components of short term components are:

(1) Seasonal Components (𝑆𝑡)

(2) Cyclical Components (𝐶𝑡)

(3) Irregular Components (𝑅𝑡)

85) Define conditional probability.

Ans:- suppose U is a finite sample space and A and B are two events in it. The probability of

occurrence of event B under the condition that A occurs is called the conditional probability

where,

𝑃(𝐵/𝐴) = 𝑃(𝐴 ∩ 𝐵)

𝑃(𝐴); 𝑃(𝐴) ≠ 0; 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦

𝑃(𝐴/𝐵) = 𝑃(𝐴 ∩ 𝐵)

𝑃(𝐵); 𝑃(𝐵) ≠ 0

86) Can mean and variance in binomial distribution be equal? Why?

Ans:- No, mean and variance in binomial distribution can not be equal. In binomial

probability distribution value of mean is always more than its variance i.e. 𝑛𝑝 > 𝑛𝑝𝑞

87) State the formula to change normal random variable X into standard normal variable𝑍.

with parameters 𝜇 and 𝜎2?

Ans:- to change normal variable X into standard normal variable Z. the formula is 𝑍 = 𝑋 − 𝜇

𝜎

88) Give the values of the constants used in probability density function of normal variable.

Ans:- The value of the constant used in probability density function of normal variable is

𝜋 = 3.1416 and 𝑒 = 2.7183

89) Define an interval.

Ans:- A set of real number between any two real number is an interval.

90) Find∫ "(𝑥) 𝑖𝑓 ∫ (𝑥) = √𝑥4

.

Ans:- ∫ (𝑥) = √𝑥4

= 𝑥1

4

∫ ′(𝑥) =1

4 𝑥

1

4−1 =1

4 𝑥

−3

4

∫ "(𝑥) = 1

4 (−

3

4) 𝑥

−3

4 −1

−3

16 𝑥

−74 =

−3

16 𝑥74

Hence, if ∫ (𝑥) = √𝑥4

𝑡ℎ𝑒𝑛 ∫ "(𝑥) =−3

16 𝑥74

91) The price of a commodity has increase by 5

4 times, then what is the index number of that

commodity?

Ans:- 𝐼𝑛𝑑𝑒𝑥𝑁𝑢𝑚𝑏𝑒𝑟 = 100 + (5

4× 100)

= 100 + 125 = 225

92) Give the definition of the coefficient of correlation.

Ans:- The numerical measure of correlation between two variables is known as correlation

coefficient and it is denoted by r.

93) Interpreter R2 = 1

Ans:- If R2 = 1 we can say that assumption or validity of regression is valid.

94) What does the error variable Ut show in any time series?

Ans:- The error variable Ut in a time series shows the influence of the other components

except trend.

95) What is an infinite sample space?

Ans:- A sample space with infinite number of elements means a sample space which is not

finite is called an infinite sample space.

96) What is random variable?

Ans:- Let U be a sample space of a random experiment. A function which associates a real

number with each outcomes of U, is called a random variable and is denoted by symbol X.

Symbolically, it is represented by X : U -> R.

97) What is the area of region bounded by normal curve and X – axis?

Ans:- The area of region bounded by normal curve and X – axis is 1.

98) Define normal variate.

Ans:- A random variable X, the mathematical form of its probability density function 𝑓(𝑥) is

as per the normal distribution is called a normal variable.

99) If N(a;0.3) = |x -5|< b then find the value of a and b.

Ans:- Comparing 𝑁 (𝑎; 0.3) = |𝑥 − 5| < 5 𝑤𝑖𝑡ℎ 𝑁(𝑎, 𝛿) = |𝑥 − 𝑎| < 𝛿; | 𝑤𝑒 𝑔𝑒𝑡 𝑎 =

5 𝑎𝑛𝑑 𝛿 = 0.3.

100) State the rule for derivative for product of two functions of x.

Ans:- If 𝑢 𝑎𝑛𝑑 𝑣 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑦 =

𝑢, 𝑣, 𝑡ℎ𝑒𝑛 𝑟𝑢𝑙𝑒 𝑓𝑜𝑟 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡𝑤𝑜 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 𝑖𝑠 𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤𝑠: 𝑑𝑦

𝑑𝑥= 𝑢

𝑑𝑣

𝑑𝑥+ 𝑣

𝑑𝑢

𝑑𝑥

101) State the main characteristics of fixed base index number.

Ans:- In fixed base method normal year is selects as the base year and price relents of

Remaining years are found out.

102) Write the formula of Karl-Pearson’s product moment method.

Ans:- r = Cov(x,y)

Sx × Sy =

𝛴(x - x)̅̅̅(y - y)̅̅ ̅

√𝛴 (x - x̅)2 ×√𝛴 (y - y̅)2

103) When two regression lines are obtained?

Ans:- When two variables are mutually dependent two regression lines are obtained.

104) State the responsible factors of random or irregular fluctuation.

Ans:- Responsible factor of random or irregular fluctuations are as under : Political

upheaval or crises. (ii) Natural disasters like earthquake, floods, draught. (iii) Random

causes like fire, accident. strike etc.

105) Define sample space.

Ans:- A set of all possible outcomes of a random experiment is called a sample space for

that random experiment. It is denoted by the symbol U or S.

106) Who has first introduced binomial probability distribution?

Ans:- James Bernoulli introduced binomial probability distribution.

107) About which perpendicular line is the normal curve symmetric?

Ans:- The normal curve is symmetric about the perpendicular line X = 𝜇.

108) What is called the value of standard normal variable z?

Ans:- It The value of standard normal variable is called standard score or Z-score

109) Express | 3x | < 1

3 in interval form.

Ans:- Comparing |3x| <1

3 with | x - 𝛿| < 𝛿 we get 𝑎 = 0 and 𝛿 =

1

9.

Interval form ∶ N(𝑎 - 𝛿; 𝑎 + 𝛿)

Putting 𝑎 = 0 and 𝛿 = 1

9 .

|3x| <1

3 = (0-

1

9; 0 +

1

9)

= ( - 1

9;

1

9)

110) State the uses of second order differentiation.

Ans:- The second order derivative can be useful in maximization of a function. This can be

applied to be minimize cost function, maximize revenue function and maximize profits

function.

111) Give the definition of index number.

Ans:- The percentage relative change in the value of a variable quantity associated with an

item for given (current) period compared to its value in the base period is called an index

number.

112) State the different method of studying correlation.

Ans:- The different methods of studying correlation are : (i) Scatter diagram method (ii) Karl

Pearson's product moment =thud, and (iii) Spearman's rank method.

113) What is the regression line of Y on X?

Ans:- The regression line in which mathematical relation between dependent variable Y and

independent variable X is given, is called regression line of Y on X

114) State the important elements of a time series?

Ans:- There are two important elements of time series : (i) Time t and (ii) Variable Quantity

yt

115) Define independent events.

Ans:- A and B are any two events of a finite sample space ∪. The event B is called

independent of A if the probability of occurrence of event A does not affect the probability

of occurrence of the event B i.e. if P (A/B) = P(A) and P(B/A) = P(B). then A and B are called

the mutually independent event.

116) Give illustrations of a continuous random variable.

Ans:- Illustrations of a continuous random variable are : (1) Age of a school teacher (2)

Height of a student (3) Temperature of a place (4) Weight of an employee.

117) State the value of median in standard normal distribution

Ans:- In standard normal distribution mean (is) median (M) = Mode (Ma) = 0. Therefore,

value of median is O.

118) What is the shape of normal curve?

Ans:- The shape of normal Curve is bell shaped.

119) Express 0.09 neighborhood of 0 in interval form.

Ans:- 0.09 neighborhood of 0 in interval form is express & by (0-0.09, 0+0.09) = (-0.09.0.09).

120) How will be the second order derivative of a function at x = a if function is minimum at

x = a?

Ans:- If function is minimum at x = a the second order derivative of a function at x a will be

f "(x)> O.

121) Write the name of two Methods of assigning explicit weight.

Ans:- (i) Total expenditure method

(ii) family budget method

122) State the types of correlation.

Ans:- there are three types of correlation

1) Positive correlation

2) Negative correlation

3) Lack of correlation

123) What is linear regression model?

Ans:-when the relation of linear correlation between two variables is represented by an

equation, it is called the liner regression model.

124) Give the additive model of time variable yt.

Ans:- 𝑦𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝐶𝑡 + 𝑅𝑡 ,

𝑤ℎ𝑒𝑟𝑒 𝑦𝑡 = value of variable at time 𝑡.

𝑇𝑡 = 𝑇𝑟𝑒𝑛𝑑𝑠 𝑡.

𝑆𝑡 = 𝑠𝑒𝑎𝑠𝑜𝑛𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑡.

𝐶𝑡 = 𝐶𝑦𝑠𝑙𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑡.

𝑅𝑡 = 𝑅𝑎𝑛𝑑𝑜𝑚 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑡.

125) What is a finite sample space?

Ans:- A random sample with finite number of elements is called a finite sample space.

126) State the types of random variable.

Ans:- Random variable is of two types

1) Discrete random variable and

2) Continuous random variable.

127) Which type of values of Z are left to perpendicular line at Z = 0?

Ans:- The value of Z left to perpendicular line at Z = 0 are negative.

128) State the parameters of normal distribution.

Ans:- the parameters of normal distribution are mean 𝜇 and standard deviation 𝜎.

129) Express 0.05 neighborhood of 0 in interval form.

Ans:- Comparing 0.05, neighborhood of 0 with 𝛿 neighborhood of a, we get a = 0 and 𝛿 =

0.05

Interval form ∶ N(𝑎 - 𝛿; 𝑎 + 𝛿)

Putting 𝑎 = 0 and 𝛿 = 0.05 .

0.05 neighborhood 0 = (0 – 0.05; 0 + 0.05)

= (-0.05; 0.05)

130) How the marginal revenue can be obtained?

Ans:- marginal revenue can be obtained by taking the derivative of the revenue function

with respect to x. thus, when the product is x then margin revenue = 𝑑R

𝑑𝑥 .