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Solved by Iftikhar Ali M.Sc. Economics, MCOM Finance Lecturer Statistics, Economics, Finance & Accounting
Table of Contents
Statistics I HSSC I FBISE Solved Paper 2023, MCQS, Short Questions, Extensive Questions
MCQS
Fill the relevant bubble against each question according to curriculum:
1  The branch of statistics that is concerned with procedures for obtaining valid conditions is called:  
A)  Descriptive Statistics  B)  Inferential Statistics 
C)  Theoretical Statistics  D)  Applied Statistics 
2  Issuing a National identity card is an example of:  
A)  Census  B)  Registration 
C)  Sampling  D)  Investigation through enumerators 
3  The process of systematic arrangement of data into rows and columns is called:  
A)  Classification  B)  Tabulation 
C)  Frequency Distribution  D)  Array 
4  An Ogive is also called:  
A)  Frequency Polygon  B)  Frequency Curve 
C)  Histogram  D)  Cumulative Frequency Polygon 
5  The modal letter(s) of the word :STATISTICS” is/are:  
A)  S  B)  T 
C)  S,T  D)  I,T 
6  If X̅ =10 and Y = 2X + 5, then Y̅ = __________:  
A)  10  B)  15 
C)  20  D)  25 
7  The sum of squared deviations from mean is always:  
A)  Negative  B)  Maximum 
C)  Minimum  D)  Zero 
8  Geometric mean of 2, 4, 6, 8, 64 is:  
A)  7  B)  7.55 
C)  16.8  D)  8.5 
9  For normal distribution, approximately 68% of the values are included by an interval:  
A)  X̅±S  B)  X̅±2S 
C)  X̅±3S  D)  X̅±4S 
10  If Var(X) = 2, then Var(3X +4) = _______:  
A)  10  B)  15 
C)  18  D)  20 
11  The reversal test is satisfied by:  
A)  Laspeyre’s Index  B)  Paasche’s Index 
C)  Fisher’s Index  D)  Unweighted Index Number 
12  The index number is given by (∑pnqn/∑poqn)x100 is called:  
A)  The Laspeyre’s index  B)  The Paasche’s index 
C)  The value index  D)  Wholesale price index 
13  The price relative is the percentage ratio of current year price and:  
A)  Current year quantity  B)  Base year quantity 
C)  Current year price  D)  Base year price 
14  In method of least square, the sum of errors will be:  
A)  Less than zero  B)  Greater than zero 
C)  Zero  D)  Not equal to zero 
15  The regression line always passing through:  
A)  (a, Y̅)  B)  (b, Y̅) 
C)  (a,b)  D)  (X̅,Y̅) 
16  When two variables move in same direction, the correlation will be:  
A)  Positive  B)  Negative 
C)  Zero  D)  Neutral 
17  A decline in death rate due to advancement of Science is an example of:  
A)  Seasonal Variation  B)  Secular Variation 
C)  Cyclical Variation  D)  Random Variation 
Total Marks Sections B and C: 68
Note: Answer any fourteen parts from section “B” and any two questions from Section “C”. Write your answers neatly and legibly. Statistical table will be provided on demand.
Section B (Marks 42)
Short Questions
(i) Distinguish between primary and secondary data.
Answer
First hand, newly collected, ungrouped data is called primary data or data which is not collected by someone previously is called primary data.
Second hand, previously collected, grouped data is called secondary data or data which is collected by someone previously is called secondary data.
(ii) Differentiate between discrete variable and continuous variable.
Answer:
Discrete Variable
Discrete variable is a variable in which a data has some specific value within a given range or we can say that discrete variable has variable that is countable. For example, number of persons, number of cars, number of students etc.
Continuous Variable
Continuous variable is a variable in which a data has any value within a given range or we can say that continuous variable has variable that is measurable. For example, height of students, speed of car, length of wood etc.
(iii) Make class boundaries and find missing frequencies of the following frequency distributions.
Classes  Frequency  Cumulative Frequency 
0.7312—0.7313  5  5 
0.7314—0.7315  7  ? 
0.7316—0.7317  ?  22 
0.7318—0.7319  8  ? 
0.7320—0.7321  ?  35 
Solution:
Classes  Frequency  Cumulative Frequency 
0.7312—0.7313  5  5 
0.7314—0.7315  7  5+7=12 
0.7316—0.7317  2212=10  22 
0.7318—0.7319  8  22+8=30 
0.7320—0.7321  3530=5  35 
(iv) Calculate Arithmetic mean from the following deviations.
D=X10  5  3  0  3  6  8  10  13 
Solution:
D=X10  5  3  0  3  6  8  10  13  ∑D=32 
(v) Calculate Geometric Mean and Harmonic Mean for five values of X for the following reciprocal values:
1/X  0.2  0.1  0.05  0.04  0.025 
Solution:
1/X  X=1/1/x 
0.2  5 
0.1  10 
0.05  20 
0.04  25 
0.025  40 
∑(1/X)=0.415 
(vi) Calculate combined Arithmetic mean for the following two groups.
Boys  Girls  
Number of Persons  20  30 
Mean Weight (kg)  62.5  53.4 
Solution:
(vii) What is meant by range and semiinterquartile range?
Answer: Range and semiinterquartile range both are absolute measures of dispersion. Range calculates the dispersion of class marks and semi interquartile deviation or quartile deviation calculates the dispersion of quartiles. Formulas are given below:
(viii) If lower quartile (Q1) = 40, upper quartile (Q3) = 90 and Median = 60, Compute Quartile Coefficient of Skewness.
Solution:
(ix) Given n =5, ∑x=180, ∑x²=6660. Compute variance, Standard deviation and coefficient of variation.
Solution:
(x) Differentiate between unweighted and weighted index number.
Answer:
Unweighted Index Number
Unweighted index number is a measure of composite index number. In unweighted index number, weights or quantities are not given. Unweighted index numbers are calculated through following methods:
 Aggregative Index
 Average of Relative Method
Weighted Index
Weighted index is also a measure of composite index numbers in which weights or quantities are given. Weighted index can be calculated through number of methods given below:
 Laspeyre’s Method
 Paasche’s Method
 Fisher’s Ideal Index Method
 Marshal Edgesworth Method
 Walsh Method
(xi) Compute index number taking 2010 as base:
Year  2010  2011  2012  2013  2014  2015 
Price  10  14  15  20  25  33 
Solution:
Year  Price  Index Number taking 2010 as base 
2010  10 

2011  14 

2012  15 

2013  20 

2014  25 

2015  33 

(xii) If Laspeyre’s index number is 150 and Fisher’s index is 147, calculate the Paasche’s index number.
Solution:
Taking square root on both sides:
(xiii) Compute two regression coefficients byx and bxy of following data: n=10, ∑Dx=12, ∑Dy=5, ∑DxDy=390,∑Dx²=2830, ∑Dy²=91
Solution:
Regression Coefficient y on x:
Regression Coefficient x on y:
(iv) The two regression lines are Ŷ=25+0.83x and X̂=40+0.97y are given, identify the two regression coefficients and compute the correlation coefficient (r).
Solution:
Regression Coefficient Y on X
Regression Coefficient X on Y
Correlation Coefficient
(xv) Write down the properties of correlation coefficient (r).
Answer:
1. Symmetric Property
Correlation between X variable and Y variable and Y Variable and X variable have same meaning or:
2. Free from unit of measurement
Correlation coefficient is free from unit of measurement means if data is given in kilograms or length, the answer will not be in kilograms or length.
3. Independent to Origin and Scale
Correlation Coefficient is independent to origin and scale means:
4. Range of Correlation Coefficient
Correlation Coefficient always lies between 1 to +1 both inclusive.
5. Geometric Mean of Regression Coefficients
Correlation Coefficient is a geometric mean of two regression coefficients.
6. Relation with Covariance
If X and Y are two independent variables then Cov(X, Y) = 0 but it does not mean that if Cov (X, Y) = 0 then X and Y are definitely independent.
(xvi) If rxy =0.60, U=(X – 50)/10, V = (Y – 60)/5, then what is the value of rxy and ruv?
Answer: It is the property of correlation coefficient that it is independent to origin and scale so rxy is equals to ruv and its value is 0.60.
(xvii) Describe seasonal variation with example in time series.
Answer: Seasonal variation is a variation of time series that occurs repeatedly when season changes. For example, in summer, the demand for cold drinks increases and when winter comes it reduces and this happens again and again.
(xviii) If the least square line fitted to the data for the years 196065 (both inclusive) with the origin at the middle of 1962 and 1963 is Ŷ = 75+0.85x, the unit of X is being half year, then find the trend values for 1960 to 1965.
Solution:
Year  X  Trend Values Ŷ = 75+0.85x 
1960  2.5  Ŷ = 75+0.85(2.5)=72.875 
1961  1.5  Ŷ = 75+0.85(1.5)=73.725 
1962  0.5  Ŷ = 75+0.85(0.5)=74.575 
1963  +0.5  Ŷ = 75+0.85(0.5)=75.425 
1964  +1.5  Ŷ = 75+0.85(1.5)=76.275 
1965  +2.5  Ŷ = 75+0.85(2.5)=77.125 
(xix) Estimate the trend values by semi average method for 1970 to 1975.
Year  SemiTotal  SemiAverage 
1970  
1971  720  240 
1972  
1973  
1974  990  330 
1975 
Solution:
Year  SemiTotal  SemiAverage  Trend Values Y̑ 
1970  240 – 30 = 210  
1971  720  240  240 
1972  240 + 30 = 270  
1973  270 + 30 = 300  
1974  990  330  330 
1975  330 + 30 = 360 
Increase of trend in 3 years = 330 – 240 = 90
Increase of trend in one year = 90/3 = 30
Section C Marks 26
Note: Attempt any two questions. All questions carry equal marks.
Extensive Questions
Q.3 a. Find Arithmetic Mean, Median and Mode from the following data: (06)
Q.3 a. Find Arithmetic Mean, Median and Mode from the following data: (06)
Classes  1014  1519  2024  2529  3034  3539 
Frequency  3  5  10  12  6  4 
Solution:
Classes  Frequency  Class Boundaries  X  fx  C.F 
10—14  3  9.5–14.5  12  36  3 
15—19  5  14.5–19.5  17  85  8 
20—24  10  19.5–24.5  22  220  18 
25—29  12  24.5–29.5  27  324  30 
30—34  6  29.5–34.5  32  192  36 
35—39  4  34.5–39.5  37  148  40 
∑f =n= 40  ∑fx = 1005 
n/2=40/2=20 falls in c.f of 30 so l=24.5, h=5,f=12,n/2=20, c=18
Maximum frequency is 12 so l = 24.5, fm =12, f1=10, f2=6 and h = 5
b. Calculate Coefficient of skewness by Karl Pearson’s Methods. (07)
b. Calculate Coefficient of skewness by Karl Pearson’s Methods. (07)
Marks  20—24  25—29  30—34  35—39  40—44  45—49 
Frequency  1  4  8  11  15  6 
Solution:
Classes  Frequency  Class Boundaries  X  fx  fx² 
20—24  1  19.5–24.5  22  22  484 
25—29  4  24.5–29.5  27  108  2916 
30—34  8  29.5–34.5  32  256  8192 
35—39  11  34.5–39.5  37  407  15059 
40—44  15  39.5–44.5  42  630  26460 
45—49  6  44.5–49.5  47  282  13254 
∑f =n= 45  ∑fx = 1705  ∑fx² = 66365 
Maximum frequency is 15 so l = 39.5, fm =15, f1=11, f2=6 and h = 5
Q.4 a. Construct chain indices using Geometric Mean as an average (08)
Q.4 a. Construct chain indices using Geometric Mean as an average (08)
Year  Prices  
Wheat  Rice  Ghee  
1990  120  30  20 
1991  132  32  24 
1992  140  38  30 
1993  144  40  40 
1994  150  45  50 
Solution:
Year  Link Relatives  Geometric Mean  
Wheat  Rice  Ghee  
1990  120  30  20 

1991  (132/120)100=110  (32/30)100=106.66  (24/20)100=120 

1992  (140/132)100=106.06  (38/32)100=118.75  (30/24)100=125 

1993  (144/140)100=102.85  (40/38)100=105.26  (40/30)100=133.33 

1994  (150/144)100=104.16  (45/40)100=112.5  (50/40)100=125 

Year  Geometric Mean  Chain Index 
1990 
 41.60 
1991 


1992 


1993 


1994 


b. Compute the consumer price index number by aggregative expenditure method (05)
b. Compute the consumer price index number by aggregative expenditure method (05)
Item  Quantity Consumed in Base Year (qo)  Base year price (po)  Current year price (pn) 
Rice  60  150  200 
Wheat  80  35  40 
Pulses  10  120  150 
Ghee  20  225  250 
Sugar  15  100  110 
Solution:
Item  (qo)  (po)  (p1)  (p1qo)  (poqo) 
Rice  60  150  200  12000  9000 
Wheat  80  35  40  3200  2800 
Pulses  10  120  150  1500  1200 
Ghee  20  225  250  5000  4500 
Sugar  15  100  110  1650  1500 
∑p1qo=23350  ∑poqo=19000 
Q.5 a. Calculate regression line Y on X and Correlation Coefficient (r) from the following data: (06)
Q.5 a. Calculate regression line Y on X and Correlation Coefficient (r) from the following data: (06)
X  10  9  8  7  6  4  3 
Y  8  12  7  10  8  9  6 
Solution:
X  Y  XY  X²  Y² 
10  8  80  100  64 
9  12  108  81  144 
8  7  56  64  49 
7  10  70  49  100 
6  8  48  36  64 
4  9  36  16  81 
3  6  18  9  36 
∑X=47  ∑Y=60  ∑XY=416  ∑X²=355  ∑Y²=538 
Where
Regression Line Y on X
Ŷ=a+bx
Where
Ŷ=a+bx
Ŷ=6.32 + 0.3353x
b. Smooth the variation of following data with the help of 4years centered moving average method. (07)
b. Smooth the variation of following data with the help of 4years centered moving average method. (07)
Year  2008  2009  2010  2011  2012  2013  2014  2015 
Profit (in Rs.000)  100  120  150  160  190  210  350  415 
Solution
Year & Quarter  Values  4 Years Moving Total  2 Values Moving Total  4 Years centered Moving Average 
2008  100  
2009  120  
530  
2010  150  1150  143.75  
620  
2011  160  1330  166.25  
710  
2012  190  1620  202.5  
910  
2013  210  2075  259.375  
1165  
2014  350  
2015  415 
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