Solved Paper Statistics I FBISE 2017, Dive into a comprehensive solution guide to the FBISE Statistics I 2017!. This blog post provides detailed explanations and step-by-step solutions for key topics like measures of central tendency, dispersion, data presentation, index numbers, correlation, regression, and time series. Whether you’re preparing for exams or reinforcing your understanding, this post is tailored to simplify concepts and help you excel in statistics. This topic is equally important for the students of statistics across all the major Boards and Universities such as FBISE, BISERWP, BISELHR, MU, DU, PU, NCERT, CBSE & others & across all the statistics, business & finance disciplines.
Table of Contents
Solved Paper Statistics I FBISE 2017
MCQS
| Q.1 Circle the Correct Option i.e. A/B/C/D. Each Part Carries 1 Mark. | |||||
| (i) | Census collects the: | ||||
| A. Primary data | B. Secondary data | C. Fictitious data | D. Official data | ||
| (ii) | The word “Statistics” has been derived from the German Word: | ||||
| A. Statista | B. Status | C. Statistik | D. Statistique | ||
| (iii) | A variable which can take any possible value in an interval is called: | ||||
| A. Discrete Variable | B. Continuous Variable | C. Qualitative Variable | D. Finite Variable | ||
| (iv) | The graph of Time Series data is called: | ||||
| A. Historigram | B. Pie-Chart | C. Histogram | D. Ogive | ||
| (v) | Total of relative frequency is called: | ||||
| A. Half | B. One | C. 100 | D. Quarter | ||
| (vi) | The sum of deviation from mean is always: | ||||
| A.Least | B. Maximum | C. One | D. Zero | ||
| (vii) | Which of the following averages is not affected by extreme values: | ||||
| A.Arithmetic Mean | B. Median | C. Mode | D. G.M | ||
| (viii) | If mean of 5 values is 10, then the sum of the values will be: | ||||
| A.2 | B. 15 | C. 25 | D. 50 | ||
| (ix) | The variance of the values 7,7,7,7,7,7 is: | ||||
| A.42 | B. 7 | C. Zero | D. | ||
| (x) | If X and Y are two independent random variables, Var(x)=4 and Var(y)=9, then Var(2x + y) is: | ||||
| A.13 | B. 17 | C. 25 | D. 26 | ||
| (xi) | In a symmetrical distribution, if Q1 = 6 and Q3 = 18 then median is: | ||||
| A. 12 | B. 15 | C. 24 | D. Zero | ||
| (xii) | The empirical relationship between mean, median and mode is: Mode =: | ||||
| A. 3Mean – 2Median | B. 2Mean – 3 Median | C. 3Median – 2 Mean | D. 2Median – 3Mean | ||
| (xiii) | The link relatives are the percentage ratios of current year price and: | ||||
| A. Previous year quantity | B. base year quantity | C. Next year price | D. Previous Year Price | ||
| (xiv) | Which index number helps the Government to formulate economic policies and determine the wages of employees: | ||||
| A. Whole sale Price Index | B. Consumer Price Index | C. Quantity Index | D. Simple Index | ||
| (xv) | The dependent variable is also called: | ||||
| A. Regressor | B. Explanatory variable | C. Predictor | D. Response Variable | ||
| (xvi) | The value of correlation coefficient lies between: | ||||
| A. 0 and 1 | B. -1 and 0 | C. -1 and 1 | D. -2 and 2 | ||
| (xvii) | Increased demand of soft drink in summer and woolen clothes in winter season is: | ||||
| A. Seasonal Variation | B. Secular Variation | C. Cyclical Variation | D. Random Variation | ||
![\[ \sqrt{7}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6595968582cda7daa89a8200719d278f_l3.png "Rendered by QuickLaTeX.com")
Short Questions
(i) Differentiate between primary and secondary data.
Answer:
Primary Data
First hand, newly collected, ungrouped data is called primary data or data which is not collected by someone previously is called primary data. For example, fresh data obtaining during research survey is an example of primary data.
Secondary Data
Second hand, previously collected, grouped data is called secondary data or data which is collected by someone previously is called secondary data. For example, data obtained from college record is an example of secondary data.
(ii) Differentiate between discrete and continuous variable.
Answer:
Discrete Variable
A variable in which data has some specific values within a given range is called discrete variable. In discrete variable, data is countable. For example, Number of students in a class, Number of houses in a street, number of children in a family etc.
Continuous Variable
A variable in which data has any values within a given range is called continuous variable. In continuous variable, data is measurable. For example, age of persons, speed of car, temperature, height, etc.
(iii) In a music competition, students are asked to rate the music on five points scale A, B, C, D, E where A represents the maximum enjoyment and E represents the minimum enjoyment. The ratings are:
A, D, A, D, E, B, C, D, A, B, B, C, E, A, C, E, C, A, B, E, D, E, B, A, B, E, E, C, B, A
Construct the frequency distribution for the above ratings
Answer:
| Ratings (X) | Frequency (f) |
| A | 7 |
| B | 7 |
| C | 5 |
| D | 4 |
| E | 7 |
| ∑f = 30 |
(iv) Write down the properties of a good average.
Solution:
(i) Deviation of X values from mean is always equal to zero.
![\[ \mathbf{\sum(X -}\overline{\mathbf{X}}\mathbf{)\ = 0\ or\ \sum f(X -}\overline{\mathbf{X}}\mathbf{)\ = 0}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-d9443774d5cb7538c25af27c1b433c8b_l3.png "Rendered by QuickLaTeX.com")
(ii) Sum of squared deviation from mean is always less than the sum of squared deviation from arbitrary origin.
![\[ \mathbf{\sum(X -}\overline{\mathbf{X}}\mathbf{)\ ²\ < \ \sum(X - A)\ ²\ or\ \sum f(X -}\overline{\mathbf{X}}\mathbf{)\ ²\ < \ \sum f(X - A)\ ²}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-fc183021c7188185287db550dfa2f547_l3.png "Rendered by QuickLaTeX.com")
(iii) Combine mean of multiple distributions can be calculated through following formula:
![\[ \mathbf{\ }\overline{\mathbf{X}}\mathbf{c}\mathbf{=}\frac{\mathbf{n1}\overline{\mathbf{X}}\mathbf{1}\mathbf{+}\mathbf{n2}\overline{\mathbf{X}}\mathbf{2}\mathbf{+}\mathbf{n3}\overline{\mathbf{X}}\mathbf{3}\mathbf{+}\mathbf{\ldots\ldots nk}\overline{\mathbf{X}}\mathbf{k}}{\mathbf{n1}\mathbf{+}\mathbf{n2}\mathbf{+}\mathbf{n3}\mathbf{+}\mathbf{\ldots\ldots nk}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-9b07cc249b5f25a4e0f7dd711602fa92_l3.png "Rendered by QuickLaTeX.com")
(v) The average marks obtained by three sections of first year class are given below: Find the combine mean of the class.
| Sections | Number of students | Means |
| A | 45 | 68 |
| B | 42 | 58 |
| C | 38 | 52 |
Solution:
![\[ \overline{\mathbf{X}}\mathbf{c = \ }\frac{\mathbf{n}\mathbf{1\ }\overline{\mathbf{X}}\mathbf{1 + n}\mathbf{2}\overline{\mathbf{X}}\mathbf{2 + n}\mathbf{3}\overline{\mathbf{X}}\mathbf{3}}{\mathbf{n}\mathbf{1 + n}\mathbf{2 + n}\mathbf{3}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-3aae25958faccd6898f5dd2f0281d086_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{c = \ }\frac{\left( \mathbf{45} \right)\left( \mathbf{68} \right)\mathbf{+}\left( \mathbf{42} \right)\left( \mathbf{58} \right)\mathbf{+}\left( \mathbf{38} \right)\left( \mathbf{52} \right)}{\mathbf{45 + 42 + 38}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-170e9607e8413d8347d57b85e67c1761_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{c =}\frac{\mathbf{3060 + 2436 + 1976}}{\mathbf{125}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-0c142f4d83f609093eaf6f3f771d18d6_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{c =}\frac{\mathbf{7472}}{\mathbf{125}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-aebf1f17f98646b4768f75ab9a9009d8_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{c = 59.776}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-7aef1949ea11097e8ee649f8967c15f5_l3.png "Rendered by QuickLaTeX.com")
(vi) For a frequency distribution of X: D=X-40, ∑fD=150, ∑f=50. Calculate arithmetic mean.
Answer:
Data
A = 40, ∑f = 50, ∑fD = 150
![\[ \overline{\mathbf{X}}\mathbf{= A +}\frac{\mathbf{\sum fD}}{\mathbf{\sum f}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-523f13fb98e2a609aea807ca531086e1_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{= 40 +}\frac{\mathbf{150}}{\mathbf{50}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6bcd43af10c83526faca1a0758d606d5_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{X}}\mathbf{= 43}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-7a6257c7740850fe879efeff930a8825_l3.png "Rendered by QuickLaTeX.com")
(vii) A variable Y is determined from a variable X by an equation, Y = 10 – 4X and X = -3, -2, -1, 0, 1, 2, 3, 4, 5. Find Ȳ and show that Ȳ = 10 – 4X̄
Solution
| X | Y = 10 – 4X | Y |
| -3 | 10 – 4(-3) = 22 | 22 |
| -2 | 10 – 4(-2) = 18 | 18 |
| -1 | 10 – 4(-1) = 14 | 14 |
| 0 | 10 – 4(0) = 10 | 10 |
| 1 | 10 – 4(1) = 6 | 6 |
| 2 | 10 – 4(2) = 2 | 2 |
| 3 | 10 – 4(3) = -2 | -2 |
| 4 | 10 – 4(4) = -6 | -6 |
| 5 | 10 – 4(5) = -10 | -10 |
| ∑X = 9 | ∑Y = 54 |
![\[ \overline{\mathbf{X}}\mathbf{=}\frac{\mathbf{\sum X}}{\mathbf{n}}\mathbf{=}\frac{\mathbf{9}}{\mathbf{9}}\mathbf{= 1}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-2b32f616acf2cc55702e07847f9e1243_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{Y}}\mathbf{=}\frac{\mathbf{\sum Y}}{\mathbf{n}}\mathbf{=}\frac{\mathbf{54}}{\mathbf{9}}\mathbf{= 6}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-93d4401da994d658f9651b6fe2765291_l3.png "Rendered by QuickLaTeX.com")
![\[ \overline{\mathbf{Y}}\mathbf{\ = \ 10\ -\ 4}\overline{\mathbf{X}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-c87effdbc6e707054935c659a2af157f_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{6\ = \ 10\ -\ 4(1)}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-aa5797555fb7dd5c2a46d37696e16804_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{6\ = 6}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-72d49963d9b25515ed50cff78d7990e4_l3.png "Rendered by QuickLaTeX.com")
(viii) Define Mean deviation and variance.
Solution
Dispersion is a measure of scatteredness or dispersement of the data. Broadly there are two types of dispersion one is called absolute dispersion which measures the dispersion in absolute terms and another is relative dispersion which measures the dispersion in relative terms.
Mean deviation and standard deviation both are absolute measures of dispersion.
Formulas for Mean Deviation from mean and variance are given below:
![\[ \mathbf{M.D\ from\ Mean =}\frac{\mathbf{\sum}\left| \mathbf{X -}\overline{\mathbf{X}} \right|}{\mathbf{n}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-25674ffb0c873d507e77007d1f80d2ac_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Variance = \ }\frac{\mathbf{\sum}\mathbf{X}^{\mathbf{2}}}{\mathbf{n}}\mathbf{-}\left( \frac{\mathbf{\sum X}}{\mathbf{n}} \right)^{\mathbf{2}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6ff4966d3f87cc8998719dd1fe45d2fd_l3.png "Rendered by QuickLaTeX.com")
(ix)For a frequency distribution of X, it is given that Mean = 50, Mode = 45 and variance = 64. Find coefficient of variation and coefficient of skewness
Solution:
Data
Mean = 50, Mode = 45, Variance = 64, Standard Deviation = 8
Coefficient of Variation
![\[ \mathbf{C.V = \ }\frac{\mathbf{S}}{\overline{\mathbf{X}}}\mathbf{\times 100}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-1032eb2297a24cc9004660baf3053dae_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{C.V =}\frac{\mathbf{8}}{\mathbf{50}}\mathbf{\times 100 = 16}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-b100505024471a56023f78081cabaeae_l3.png "Rendered by QuickLaTeX.com")
Karl Pearson Coefficient of Skewness
![\[ \mathbf{C.\ of\ Skewness = \ }\frac{\mathbf{Mean - Mode\ }}{\mathbf{S.D}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-b62a3f4ecee8847f68104df3053535c7_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{C.\ of\ Skewness = \ }\frac{\mathbf{50 - 45\ }}{\mathbf{16}}\mathbf{= 0.625}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-52e6a1b59adfc8b1e1fd2e44d5004dc9_l3.png "Rendered by QuickLaTeX.com")
(x) If Mean = 75, Mode= 70, using empirical relation, find the value of Median.
Answer:
Data
Mean = 75, Mode = 70
Mode = 3Median – 2Mean
70 = 3Median – 2(75)
70 = 3Median – 150
3Median = 70 + 150
3Median = 120
Median = 120/3
Median = 40
(xi) Differentiate between fixed and chain base method.
Answer:
In fixed base method, base period remains fixed whereas in chain base method base period does not remain fixed. On the other hand, in fixed base method, we calculate price relative whereas in chain base method, we calculate link relative and then we calculate chain index. Both methods can be used in simple and composite index. Formulas for Price and Link relatives are given below:
![\[ \mathbf{Price\ Relative = \ }\frac{\mathbf{Pn}}{\mathbf{Po}}\mathbf{\ \times \ 100}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-3b2b2017275f03f3ad1ac2bf94b820b9_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Link\ Relative = \ }\frac{\mathbf{Pn}}{\mathbf{Pn - 1}}\mathbf{\ \times \ 100}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8e7ef02d8c785109cfdefdbce0d7adad_l3.png "Rendered by QuickLaTeX.com")
(xii) Compute the chain indices from the following:
| Year | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 |
| Price | 10 | 12 | 15 | 20 | 25 | 30 |
Answer:
| Year | Price | Link Relative | Chain Index |
| 1990 | 10 | 100 | 100 |
| 1991 | 12 | (12/10) x 100 = 120 | (100 x 120)/100 = 120 |
| 1992 | 15 | (15/12) x 100 = 125 | (120 x 125)/100 = 150 |
| 1993 | 20 | (20/15) x 100 = 133.34 | (150 x 133.34)/100 = 200 |
| 1994 | 25 | (25/20) x 100 = 125 | (200 x 125)/100 = 250 |
| 1995 | 30 | (30/25) x 100 = 120 | (250 x 120)/100 = 300 |
(xiii) Define Consumer Price Index and write down the major groups included in CPI.
Solution
Consumer Price Index
Consumer Price Index numbers are used to measure the changes in the prices paid by the consumer for purchasing a specified “basket” of goods and services during the current year as compared to the base year.
Major Groups Included in CPI
Food & Beverages
Housing
Transportation
Healthcare
Education
Energy
Clothing etc.
(xiv) If Laspeyre’s index is 120 and Paasche’s index is 130, then find Fisher’s index number.
Solution
![\[ \mathbf{Fishe}\mathbf{r}^{\mathbf{'}}\mathbf{s\ Ideal\ Index =}\sqrt{\mathbf{Las.\ Index \times Paas.\ Index\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f5fffb91bb659946b6d1809f9576e9fc_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Fishe}\mathbf{r}^{\mathbf{'}}\mathbf{s\ Ideal\ Index =}\sqrt{\mathbf{120 \times 130\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-a360236e611b5b5924a06b80e1b65958_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Fishe}\mathbf{r}^{\mathbf{'}}\mathbf{s\ Ideal\ Index = 124.89}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-44b4455cb442eb2c3bd40bf6c990d784_l3.png "Rendered by QuickLaTeX.com")
(xv) Differentiate between regression and correlation
Answer:
Correlation
Correlation or Correlation coefficient is a statistical measure that calculates the quantitative relationship between two variables. It is denoted by (r). It is always ranges between -1 to +1 both inclusive. It has five different types namely (i) Positive (ii) Perfectly positive (iii) Negative (iv) Perfectly negative and (v) Zero.
Regression
Regression or regression coefficient is also a statistical measure which calculates the strength of relationship between two or more than two variables in which one is dependent and all other variables are considered as independent variables. Two variable model is called simple regression and model which has more than two variables is called multiple regression. Regression is very important tool used in forecasting or trend analysis.
(xvi) It is given that: Syx=32, Sx = 2.4, Sy = 25, X̅=155, Y̅=7, n = 10. Calculate Regression Coefficients byx and bxy.
Solution:
Data
Here
Covariance of X and Y = Syx=32
Standard Deviation of X variable = Sx = 2.4
Variance of X Variable S²x = 5.76
Standard Deviation of Y variable = Sx = 25
Variance of Y Variable S²y = 625
Mean of X Variable = X̅=155
Mean of Y Variable = Y̅=7 and n = 10
Regression Coefficient of Y on X
![\[ \mathbf{b}{\mathbf{yx}}\mathbf{=}\frac{\mathbf{S}{\mathbf{yx}}}{\mathbf{S}_{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{32}}{\mathbf{5.76}}\mathbf{= 5.56}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-493b9268c2ce86b57fc6591933d2d71c_l3.png "Rendered by QuickLaTeX.com")
Regression Coefficient of X on Y
![\[ \mathbf{b}{\mathbf{yx}}\mathbf{=}\frac{\mathbf{S}{\mathbf{yx}}}{\mathbf{S}_{\mathbf{y}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{32}}{\mathbf{625}}\mathbf{= 0.0512}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-0f73306149c45835cca3279b3c6e4703_l3.png "Rendered by QuickLaTeX.com")
(xvii) Find the correlation coefficients (r) from the regression coefficients. (i) 0.85 and 0.6 (ii) -0.96 and -0.55
Solution:
![\[ \left( \mathbf{a} \right)\mathbf{r =}\sqrt{\mathbf{b}_{\mathbf{yx}}\mathbf{\times}\mathbf{b}_{\mathbf{xy}}\mathbf{\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f86ad90c05317113a7009d890df84ca6_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r =}\sqrt{\mathbf{0.85 \times 0.6\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-80f8b302a96726a528e85b733a36e3ee_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r =}\sqrt{\mathbf{051\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-d31e767be4eb67370650a72e2f12e6f0_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = 0.714}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6be5554604a6748b30f542355de4b207_l3.png "Rendered by QuickLaTeX.com")
![\[ \left( \mathbf{b} \right)\mathbf{r = -}\sqrt{\mathbf{b}_{\mathbf{yx}}\mathbf{\times}\mathbf{b}_{\mathbf{xy}}\mathbf{\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-329678c0bccf30504821f88ab1d27d50_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = -}\sqrt{\mathbf{( - 0.96)( - 0.55)\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-b3b7ef69c3d0ab5ed8c6b2a4d11b8108_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = -}\sqrt{\mathbf{0.528\ }}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-93b2f5e76858fd03120c579a5863afa0_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = - 0.7266}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-d6a992da015f1f7fb484577d87097ea2_l3.png "Rendered by QuickLaTeX.com")
(xviii) What are the different components of a time series?
Answer
The factors that are responsible to bring about changes in a time series, also called the components of time series, are as follows:
- Secular Trend (T)
- Seasonal Movements (S)
- Cyclical Movements (C)
- Irregular Fluctuations or movements (I)
(xix) Calculate three years moving average for the following time series:
| Year | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 |
| Sale | 100 | 140 | 168 | 120 | 200 | 210 | 170 | 220 |
Answer:
| Year | Sale | 3 Year M.T | 3 Year M.A |
| 1980 | 100 | – | – |
| 1981 | 140 | 408 | 136 |
| 1982 | 168 | 428 | 142.67 |
| 1983 | 120 | 488 | 162.67 |
| 1984 | 200 | 530 | 176.67 |
| 1985 | 210 | 580 | 193.34 |
| 1986 | 170 | 600 | 200 |
| 1987 | 220 | – | – |
Extensive Questions
Q.3 (a) Find Arithmetic Mean, Median, Mode Q1 and Q3 of the following frequency distribution:
| Max: Load (Short tons) | No of cables |
| 118—126 | 2 |
| 127—135 | 5 |
| 136—144 | 12 |
| 145—153 | 17 |
| 154—162 | 14 |
| 163—171 | 6 |
| 172—180 | 3 |
Solution:
| Max: Load (Short tons) | No of cables (f) | C.B | X | fx | C.F |
| 118—126 | 2 | 117.5—126.5 | 122 | 244 | 2 |
| 127—135 | 5 | 126.5—135.5 | 131 | 655 | 7 |
| 136—144 | 12 | 135.5—144.5 | 140 | 1680 | 19 |
| 145—153 | 17 | 144.5—153.5 | 149 | 2533 | 36 |
| 154—162 | 14 | 153.5—162.5 | 158 | 2212 | 50 |
| 163—171 | 6 | 162.5—171.5 | 167 | 1002 | 56 |
| 172—180 | 3 | 171.5—180.5 | 176 | 528 | 59 |
| ∑f=n=59 | ∑fx =8854 |
Mean
![\[ \mathbf{Mean\ }\overline{\mathbf{X}}\mathbf{=}\frac{\mathbf{\sum X}}{\mathbf{\sum f}}\mathbf{=}\frac{\mathbf{8854}}{\mathbf{59}}\mathbf{= 150.06}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-7b41f2980a785231f85a651af2337c1c_l3.png "Rendered by QuickLaTeX.com")
Median
∑f or n = 59, n/2 = 59/2 = 29.5 falls in C.F of 36 so l = 144.5, f = 17, h = 9 & C = 19
![\[ \widetilde{\mathbf{X}}\mathbf{= l +}\frac{\mathbf{h}}{\mathbf{f}}\left( \frac{\mathbf{n}}{\mathbf{2}}\mathbf{- C} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8fdb00fc266327c377c2ef8f3ec13cf3_l3.png "Rendered by QuickLaTeX.com")
![\[ \widetilde{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{9}}{\mathbf{17}}\left( \mathbf{29.5 - 19} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8c44a7b616af447add7d7c6e78d99639_l3.png "Rendered by QuickLaTeX.com")
![\[ \widetilde{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{9}}{\mathbf{17}}\left( \mathbf{10.5} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-a9bf2d4ded0c8aa4294889adea3046fe_l3.png "Rendered by QuickLaTeX.com")
![\[ \widetilde{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{94.5}}{\mathbf{17}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-b08f24aa91aa9fe66bc0abcd844d5733_l3.png "Rendered by QuickLaTeX.com")
![\[ \widetilde{\mathbf{X}}\mathbf{= 150.05}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-13a3f25bbc6780e42185131190d8df24_l3.png "Rendered by QuickLaTeX.com")
Mode
Maximum frequency is 17 so fm = 17, f1 = 12, f2 = 14, h = 9 and l = 144.5
![\[ \widehat{\mathbf{X}}\mathbf{= l +}\frac{\mathbf{f}_{\mathbf{m}}\mathbf{-}\mathbf{f}_{\mathbf{1}}}{\left( \mathbf{f}_{\mathbf{m}}\mathbf{-}\mathbf{f}_{\mathbf{1}} \right)\mathbf{+ (}\mathbf{f}_{\mathbf{m}}\mathbf{-}\mathbf{f}_{\mathbf{2}}\mathbf{)}}\mathbf{\times h}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-ed49c64a7fbabfe03b83de91bbe2a3af_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{17 - 12}}{\left( \mathbf{17 - 12} \right)\mathbf{+ (17 - 14)}}\mathbf{\times 9}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-403b6de4da499be86adc814994ab844b_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{5}}{\mathbf{5 + 3}}\mathbf{\times 9}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-5885603cdbf70b502dd606d9237bac50_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{X}}\mathbf{= 144.5 +}\frac{\mathbf{45}}{\mathbf{8}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-5901301ce42cc91519c8d995131b2ecf_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{X}}\mathbf{= 5.625}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-70668ac36e5fe924b538c9646f88eda2_l3.png "Rendered by QuickLaTeX.com")
Quartile 1 Q1
∑f or n = 59, 1n/4 = 59/4 = 14.75 falls in C.F of 19 so l = 135.5, f = 12, h = 9 & C = 7
![\[ \mathbf{Q}_{\mathbf{1}}\mathbf{= l +}\frac{\mathbf{h}}{\mathbf{f}}\left( \frac{\mathbf{1}\mathbf{n}}{\mathbf{4}}\mathbf{- C} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f6a77fc07986f195f4d748c5f3aecad1_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{1}}\mathbf{= 135.5 +}\frac{\mathbf{9}}{\mathbf{12}}\left( \mathbf{14.75 - 7} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f707128d7dc30a251cb4a9a158bcbd70_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{1}}\mathbf{= 135.5 +}\frac{\mathbf{9}}{\mathbf{12}}\left( \mathbf{7.75} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f44180f0737417f44e8aee33c542d54d_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{1}}\mathbf{= 135.5 +}\frac{\mathbf{69.75}}{\mathbf{12}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-67af66bc90c3176390052fb4f338531d_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{1}}\mathbf{= 141.3125}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f58d498db4d5925d4d8b9d0c8c046450_l3.png "Rendered by QuickLaTeX.com")
Quartile 3 Q3
∑f or n = 59, 3n/4 = [(3)(59)]/4 = 44.25 falls in C.F of 50 so l = 153.5, f = 14, h = 9 & C = 36
![\[ \mathbf{Q}_{\mathbf{3}}\mathbf{= l +}\frac{\mathbf{h}}{\mathbf{f}}\left( \frac{\mathbf{1}\mathbf{n}}{\mathbf{4}}\mathbf{- C} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-eb936c70c43adff782e404b975ab58af_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{3}}\mathbf{= 153.5 +}\frac{\mathbf{9}}{\mathbf{14}}\left( \mathbf{44.25 - 36} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-7ca4bc4acfcfdbb64eb8308a88092cec_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{3}}\mathbf{= 153.5 +}\frac{\mathbf{9}}{\mathbf{14}}\left( \mathbf{8.25} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-9aeade7b6412aedaf0fca667bc94f175_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{3}}\mathbf{= 153.5 +}\frac{\mathbf{74.25}}{\mathbf{14}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-4cb81bae944e99c9eb2ce5a4e06b2034_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Q}_{\mathbf{3}}\mathbf{= 158.80}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-1c53f97e05d09d54da44b18ba4a44ff9_l3.png "Rendered by QuickLaTeX.com")
(b) Calculate variance, standard deviation and coefficient of variation:
| Daily Wages | Frequency |
| 1—3 | 2 |
| 3—5 | 4 |
| 5—7 | 10 |
| 7—9 | 3 |
| 9—11 | 1 |
Solution
| Daily Wages | Frequency (f) | X | fx | fx² |
| 1—3 | 2 | 2 | 4 | 8 |
| 3—5 | 4 | 4 | 16 | 64 |
| 5—7 | 10 | 6 | 60 | 360 |
| 7—9 | 3 | 8 | 24 | 192 |
| 9—11 | 1 | 10 | 10 | 100 |
| ∑f = 20 | ∑fx =114 | ∑fx² =724 |
![\[ \mathbf{Variance\ }\mathbf{S}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{\sum f}\mathbf{x}^{\mathbf{2}}}{\mathbf{\sum f}}\mathbf{-}\left( \frac{\mathbf{\sum fx}}{\mathbf{\sum f}} \right)^{\mathbf{2}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-12b3908b9570a7b0548d33fd67f0e4b5_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Variance\ }\mathbf{S}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{724}}{\mathbf{20}}\mathbf{-}\left( \frac{\mathbf{114}}{\mathbf{20}} \right)^{\mathbf{2}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8e45e9362dc263937f1601be7705c645_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Variance\ }\mathbf{S}^{\mathbf{2}}\mathbf{= 36.2 - 32.49}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-ac4b9039d67b1e3ae4bea5a627056035_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Variance\ }\mathbf{S}^{\mathbf{2}}\mathbf{= 3.71}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8f53ffcf5240920bce38235f45855fca_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{S.D\ S =}\sqrt{\frac{\mathbf{\sum f}\mathbf{x}^{\mathbf{2}}}{\mathbf{\sum f}}\mathbf{-}\left( \frac{\mathbf{\sum fx}}{\mathbf{\sum f}} \right)^{\mathbf{2}}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-1cf8ac64dbaa31fb81b51278b2a8e512_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{S.D\ S =}\sqrt{\mathbf{3.71}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-d2b61e67c4cd577cf0efaa27f914d0c2_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{S.D\ S = 1.92}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-13b00544097f212dd8a89cefb61abb46_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Mean\ }\overline{\mathbf{X}}\mathbf{=}\frac{\mathbf{\sum fx}}{\mathbf{\sum f}}\mathbf{=}\frac{\mathbf{114}}{\mathbf{20}}\mathbf{= 5.7\ }\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-77b28a35e1cdeccc54a8309ee17c86a3_l3.png "Rendered by QuickLaTeX.com")
Coefficient of Variation
![\[ \mathbf{C.V =}\frac{\mathbf{1.92}}{\mathbf{5.7}}\mathbf{\times 100 = 33.68}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-7a0d447e6e1f1be0d69d39bb0df540f6_l3.png "Rendered by QuickLaTeX.com")
Q.4 (a) Compute Chain index from the following price relatives using (i) Mean (ii) geometric mean as an average:
| Year | Commodities | |||
| A | B | C | D | |
| 2000 | 81 | 77 | 119 | 55 |
| 2001 | 62 | 54 | 128 | 52 |
| 2002 | 104 | 87 | 111 | 100 |
| 2003 | 93 | 75 | 154 | 96 |
| 2004 | 60 | 43 | 165 | 88 |
Solution
| Year | Link Relatives | |||
| A | B | C | D | |
| 2000 | 81 | 77 | 119 | 55 |
| 2001 | (62/81)×100=76.54 | (54/77)×100=70.12 | (384/330)×100=116.36 | (52/55)×100= 94.54 |
| 2002 | (104/62)×100=167.74 | (87/54)×100=161.12 | (333/384) × 100=86.71 | (100/52)×100= 192.30 |
| 2003 | (93/104) × 100=89.42 | (75/87) × 100=86.20 | (462/333)×100=138.73 | (96/100)×100= 96 |
| 2004 | (60/93) × 100=64.51 | (43/75) × 100=57.33 | (495/462)×100=107.14 | (88/96)×100= 91.67 |
- Chain Index Using Mean as an average
| Year | X̅ = ∑X/n | |
| 2000 | 332/4 =83 | 83 |
| 2001 | 357.56/4 = 89.39 | |
| 2002 | 607.87/4 = 151.96 | |
| 2003 | 410.35/4 = 102.58 | |
| 2004 | 320.65/4= 80.1625 | |
![\[ \frac{\mathbf{83}\mathbf{\ \times \ 8}\mathbf{9.39}}{\mathbf{100}}\mathbf{=}\mathbf{74.1937}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6c692afef6fea9e00a07e76e64d525f6_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{74.1937}\mathbf{\ \times \ }\mathbf{151.96}}{\mathbf{100}}\mathbf{=}\mathbf{112.74}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-97b744f4f645a566bad5b65592d97a09_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{112.74}\mathbf{\ \times \ 102.}\mathbf{58}}{\mathbf{100}}\mathbf{=}\mathbf{115.64}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-90ade34b712139eeb2709803a71e7c55_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{115.64}\mathbf{\ \times \ }\mathbf{80.1625}}{\mathbf{100}}\mathbf{=}\mathbf{92.699}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-3eb3fc198a30c3f5012e14296bcf88e0_l3.png "Rendered by QuickLaTeX.com")
- Chain Index Using Geometric Mean as an average
| Year | | |
| 2000 | | 79.93 |
| 2001 | | |
| 2002 | | |
| 2003 | | |
| 2004 | | |
![\[ {\mathbf{G.M(X}\mathbf{1.X}\mathbf{2.X}\mathbf{3)}}^{\mathbf{1/3}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-b539988fc171abf606ff2271af092c18_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{C.I = \ }\frac{\mathbf{Chai}\mathbf{n}{\mathbf{n - 1}}\mathbf{\ \times \ Averag}\mathbf{e}{\mathbf{n}}\mathbf{\ L.R}}{\mathbf{100}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-957b0c70bd75aa6d8a090f00d23ef008_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{(81 \times 77 \times 119 \times 55)}^{\mathbf{1/4}}\mathbf{= 79.93}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-a0c039399223b0ef2e57dc43d86f03dd_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{(76.54 \times 70.12 \times 116.36 \times 94.54)}^{\mathbf{1/4}}\mathbf{= 87.65}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-420a6b04c1fb944bc13c0b0a8f9c6b5e_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{79.93\ \times \ 87.65}}{\mathbf{100}}\mathbf{= 70.05}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-97a8762a3009681582437e1021ba9343_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{(167.74 \times 161.12 \times 86.71 \times 192.30)}^{\mathbf{1/4}}\mathbf{= 145.69}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-87c28a34828ff162d510177832889b00_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{70.05\ \times \ 145.69}}{\mathbf{100}}\mathbf{= 102.05}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-5d4a6936407b029544c8db4d784a3c6c_l3.png "Rendered by QuickLaTeX.com")
![\[ {\mathbf{(89.42 \times 86.20 \times 138.73}\mathbf{\times 96}\mathbf{)}}^{\mathbf{1/}\mathbf{4}}\mathbf{= 10}\mathbf{0.65}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-dd53a03e058e02ef215af1090cbd8786_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{102.05}\mathbf{\ \times \ 1}\mathbf{00.65}}{\mathbf{100}}\mathbf{=}\mathbf{102.71}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-2aa289b6ae1efb4dbd15f0896a849346_l3.png "Rendered by QuickLaTeX.com")
![\[ {\mathbf{(64.51 \times 57.33 \times 107.14}\mathbf{\times 91.67}\mathbf{)}}^{\mathbf{1/}\mathbf{4}}\mathbf{= 7}\mathbf{7.63}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-e30784a58c12254705b581cf64899c14_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{\mathbf{102.71}\mathbf{\ \times \ 7}\mathbf{7.63}}{\mathbf{100}}\mathbf{=}\mathbf{79.73}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-66cb8cad5ce5102b3601b1b26c879f34_l3.png "Rendered by QuickLaTeX.com")
(b) An inquiry into budgets of the middle class families in a city for year 1989 – 1990 was conducted. Construct Consumer Price Index.
The following price relatives are given:
| Expenses | Food | Rent | Clothing | Fuel | Misc. |
| Weights (W) | 35% | 15% | 20% | 10% | 20% |
| Price Relative (I) | 116 | 120 | 125 | 125 | 150 |
Solution:
| Expenses | Weights (W) | Price Relative (I) | WI |
| Food | 35 | 116 | 4060 |
| Rent | 15 | 120 | 1800 |
| Clothing | 20 | 125 | 2500 |
| Fuel | 10 | 125 | 1250 |
| Misc. | 20 | 150 | 3000 |
| ∑W = 100 | ∑WI = 12610 |
Method to Calculate
Household Budget method or Family Budget Method
![\[ \mathbf{P}_{\mathbf{on}}\mathbf{=}\frac{\mathbf{\sum}\left( \frac{\mathbf{Pn}}{\mathbf{Po}} \right)\mathbf{Poqo}}{\mathbf{\sum Poqo}}\mathbf{\times 100 =}\frac{\mathbf{\sum IW}}{\mathbf{\sum W}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-6084156c4c39ae53ae7e5eca056f91cb_l3.png "Rendered by QuickLaTeX.com")
Where Price Relative I equals to:
![\[ \frac{\mathbf{Pn}}{\mathbf{Po}}\mathbf{\times 100,\ W = Poqo}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f0f1de8a3f2492ee4960861627118037_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{P}_{\mathbf{on}}\mathbf{=}\frac{\mathbf{\sum}\mathbf{WI}}{\mathbf{\sum W}}\mathbf{=}\frac{\mathbf{12610}}{\mathbf{100}}\mathbf{= 126.1}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-c67e35fe0ae472ce6110fb62c8b6a4b2_l3.png "Rendered by QuickLaTeX.com")
Q.5 (a) The following data is obtained in a study on the number of absentees (X) and the final Marks (Y) of seven students from a class. (i) Compute Correlation coefficient (r) (ii) Obtain regression line Yon X and estimate final marks when there are 20 absentees.
| X | 6 | 2 | 15 | 9 | 12 | 5 | 8 |
| Y | 85 | 86 | 43 | 74 | 58 | 90 | 78 |
Solution:
| X | Y | XY | X² | Y² |
| 6 | 85 | 510 | 36 | 7225 |
| 2 | 86 | 172 | 4 | 7396 |
| 15 | 43 | 645 | 225 | 1849 |
| 9 | 74 | 666 | 81 | 5476 |
| 12 | 58 | 696 | 144 | 3364 |
| 5 | 90 | 450 | 25 | 8100 |
| 8 | 78 | 624 | 64 | 6084 |
| ∑X = 57 | ∑Y = 514 | ∑XY = 3763 | ∑X² = 579 | ∑Y² = 39494 |
Solution (i)
Correlation Coefficient (r)
Regression Coefficient X on Y
![\[ \mathbf{d}\mathbf{= \ }\frac{\mathbf{\sum}\mathbf{xy - n}\overline{\mathbf{X}}\overline{\mathbf{Y}}}{\mathbf{\sum}\mathbf{Y}^{\mathbf{2}}\mathbf{- n}\mathbf{(}\overline{\mathbf{Y}}\mathbf{)}\mathbf{²}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-1389873caf7ed002b9bd5552bcf04bad_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{d}\mathbf{= \ }\frac{\mathbf{3763 -}\left( \mathbf{7} \right)\left( \mathbf{8.14} \right)\left( \mathbf{73.42} \right)}{\mathbf{39494}\mathbf{- 7}\left( \mathbf{73.42} \right)^{\mathbf{2}}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-af0ea26cf1371df7aaab92b2e12637fb_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{d}\mathbf{= \ }\frac{\mathbf{- 420.4716}}{\mathbf{1}\mathbf{760.5252}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-380089baac31ad94a506bace3c8fe8d6_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{d}\mathbf{=}\mathbf{-}\mathbf{0.2388}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-2f23d15c2c0e50addfb8bd8fff01068a_l3.png "Rendered by QuickLaTeX.com")
Regression Coefficient Y on X (b) is calculated in (ii)
![\[ \mathbf{r = \pm}\sqrt{\mathbf{b \times d}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-e6494a825b93db308d9ab80effb7b82f_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = -}\sqrt{\mathbf{( - 3.65)( - 0.2388)}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-12ce0317d6013e7d19e92fccf7d17004_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = -}\sqrt{\mathbf{0.87162}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-46a265369bfe75cccbece29b195b08e3_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{r = -}\mathbf{0.9336\ Negative\ Relation}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-71f4f7cfeb442da1d4c0a30385022e40_l3.png "Rendered by QuickLaTeX.com")
Solution (ii)
![\[ \mathbf{a}\mathbf{=}\overline{\mathbf{y}}\mathbf{- b}\overline{\mathbf{X}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-ce6d2ee229e792c56e2eaa9808fe5abc_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{b}\mathbf{= \ }\frac{\mathbf{\sum}\mathbf{xy - n}\overline{\mathbf{X}}\overline{\mathbf{Y}}}{\mathbf{\sum}\mathbf{X}^{\mathbf{2}}\mathbf{- n}\mathbf{(}\overline{\mathbf{X}}\mathbf{)}\mathbf{²}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-3625f18faa24300402ae4529823b026d_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Mean}\overline{\mathbf{X}}\mathbf{=}\frac{\mathbf{\sum}\mathbf{X}}{\mathbf{n}}\mathbf{=}\frac{\mathbf{57}}{\mathbf{7}}\mathbf{= 8.14}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-8287e2f619b96ac9dd9cbe00983a1f5d_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{Mean}\overline{\mathbf{Y}}\mathbf{=}\frac{\mathbf{\sum}\mathbf{Y}}{\mathbf{n}}\mathbf{=}\frac{\mathbf{514}}{\mathbf{7}}\mathbf{=}\mathbf{73.42}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-2c56779c3aa39f386f7d4367643107ea_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{b}\mathbf{= \ }\frac{\mathbf{\sum}\mathbf{xy - n}\overline{\mathbf{X}}\overline{\mathbf{Y}}}{\mathbf{\sum}\mathbf{X}^{\mathbf{2}}\mathbf{- n}\left( \overline{\mathbf{X}} \right)^{\mathbf{2}}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-a3f054e5c6eb418fb0cafb572bfe637c_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{b}\mathbf{= \ }\frac{\mathbf{3763}\mathbf{-}\left( \mathbf{7} \right)\left( \mathbf{8.14} \right)\left( \mathbf{73.42} \right)}{\mathbf{579}\mathbf{-}\mathbf{7}\left( \mathbf{8.14} \right)^{\mathbf{2}}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-dd98a0d1c2aaa7f12276fc0e1ab754d1_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{b}\mathbf{= \ }\frac{\mathbf{- 420.4716}}{\mathbf{115.1828}}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-805d9464fdda4b92c8dc71ce09500777_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{b}\mathbf{=}\mathbf{-}\mathbf{3.65}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-75b329e23d35ac187cfcc784616aa246_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{a}\mathbf{=}\mathbf{73.42}\mathbf{-}\left( \mathbf{- 3.65} \right)\left( \mathbf{8.14} \right)\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-5945a15cc35ce58e023fc8f6f1d4d1a0_l3.png "Rendered by QuickLaTeX.com")
![\[ \mathbf{a}\mathbf{=}\mathbf{103.131}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-f4101207a5dfceb9dc4cc3b578df185d_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{y}}\mathbf{\ = \ a\ + \ bx\ }\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-85f148ded72c20ddc72ba9c3c0a8254d_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{y}}\mathbf{= \ }\mathbf{103.131 -}\mathbf{\ }\mathbf{3.65}\mathbf{x}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-664d006715a926a4f20632db5920b23f_l3.png "Rendered by QuickLaTeX.com")
Final marks when there are 20 absentees
![\[ \widehat{\mathbf{y}}\mathbf{= \ 103.131 - \ 3.65x}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-1bf152a4dbc9f5c9905390398b94d0f3_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{y}}\mathbf{= \ 103.131 - \ 3.65}\mathbf{(20)}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-a85cb9f1bc1011030e48b2920883ff0f_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{03.131 -}\mathbf{73}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-9fa472dbf84c691ffc6ad90bde4e97a1_l3.png "Rendered by QuickLaTeX.com")
![\[ \widehat{\mathbf{y}}\mathbf{=}\mathbf{30.131}\ \]](https://bcfeducation.com/wp-content/ql-cache/quicklatex.com-2f425849fe3bd7d1abf13a60ae7b5c58_l3.png "Rendered by QuickLaTeX.com")
Approximately 30 marks if absentees are 20
b. Find 4-qurarters centered moving averages from the following time series data:
| Years | Quarters | |||
| I | II | III | IV | |
| 2005 | 160 | 165 | 163 | 161 |
| 2006 | 170 | 167 | 172 | 171 |
| 2007 | 172 | 169 | 167 | 170 |
| 2008 | 175 | 177 | 172 | 170 |
Solution:
| Year & Quarter | Values | 4 Quarter Moving Total (C.1) | 2 Values Moving Total (C.2) | 4 Quarter centered Moving Average (C.2/8) |
| 2005 (I) | 160 | |||
| (ii) | 165 | |||
| 649 | ||||
| (iii) | 163 | 1308 | 163.5 | |
| 659 | ||||
| (iv) | 161 | 1320 | 165 | |
| 661 | ||||
| 2006 (i) | 170 | 1331 | 166.375 | |
| 670 | ||||
| (ii) | 167 | 1350 | 168.75 | |
| 680 | ||||
| (iii) | 172 | 1362 | 170.25 | |
| 682 | ||||
| (iv) | 171 | 1366 | 170.75 | |
| 684 | ||||
| 2007 (i) | 172 | 1363 | 170.375 | |
| 679 | ||||
| (ii) | 169 | 1357 | 169.625 | |
| 678 | ||||
| (iii) | 167 | 1359 | 169.875 | |
| 681 | ||||
| (iv) | 170 | 1370 | 171.25 | |
| 689 | ||||
| 2008 (i) | 175 | 1383 | 172.875 | |
| 694 | ||||
| (ii) | 177 | 1388 | 173.5 | |
| 694 | ||||
| (iii) | 172 | |||
| (iv) | 170 |
Related Articles
Statistics I Solved Paper FBISE 2008
Statistics I Solved Paper FBISE 2009
Statistics I Solved Paper FBISE 2010
Statistics I Solved Paper FBISE 2011
Solved Paper Statistics I FBISE 2012
Solved Paper Statistics I FBISE 2015 & 2015 2nd Annual
