Applied Statistics Exercise 9.1

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 9.1 Business Maths Chapter 9 Applied Statistics for your reference and study.

You can find the links to the other exercises below.

Applied Statistics Important Formulas

Solutions for Applied Statistics Exercise 9.2

Solutions for Applied Statistics Exercise 9.3

Solutions for Applied Statistics Exercise 9.4

Applied Statistics Exercise 9.1

$\text{1. Define Time series.}\\$
$\text{A Time Series consists of data arranged chronologically - Croxton & Cowden.} \\$
$\text{When quantitative data are arranged in the order of their occurrence, }\\$
$\text{the resulting series is called the Time Series - Wessel & Wallet.} \\$

$\text{2. What is the need for studying time series?} \\$
$\text{We need to study Time Series to identify the variations and try to eliminate }\\$
$\text{them and also helps us to estimate or predict the future values.} \\$

$\text{3. State the uses of time series.}\\$
$\text{1. It helps in the analysis of the past behavior.}\\$
$\text{2. It helps in forecasting and for future plans.}\\$
$\text{3. It helps in the evaluation of current achievements.} \\$
$\text{4. It helps in making comparative studies between one time period and others.} \\$

$\text{ 4. Mention the components of the time series.}\\$
$\text{There are four types of components in a time series. They are as follows;}\\$
$\text{(i) Secular Trend} \\$
$\text{(ii) Seasonal variations}\\$
$\text{(iii) Cyclic variations} \\$
$\text{(iv) Irregular variations}\\$

$\text{5. Define secular trend.} \\$
$\text{It is a general tendency of time series to increase or decrease or stagnate}\\$
$\text{during a long period of time. An upward tendency is usually observed in } \\$
$\text{population of a country, production, sales, prices in industries, income }\\$
$\text{of individuals etc., A downward tendency is observed in deaths, epidemics,} \\$
$\text{prices of electronic gadgets,water sources, mortality rate etc. It is not}\\$
$\text{necessarily that the increase or decrease should be in the same }\\$
$\text{direction throughout the given period of time.}\\$

$\text{6. Write a brief note on seasonal variations}\\$
$\text{Tendency movements are due to nature which repeat themselves periodically}\\$
$\text{in every season. These variations repeat themselves in less than one year's}\\$
$\text{time. It is measured in an interval of time. Seasonal variations may be} \\$
$\text{influenced by natural force, social customs and traditions. These variations}\\$
$\text{are the results of such factors which uniformly and regularly rise and fall}\\$
$\text{in the magnitude. For example, selling of umbrellas and raincoats in the rainy}\\$
$\text{season, sales of cool drinks in summer season, crackers in Deepawali season etc. }\\$

$\text{7. Explain cyclic variations.} \\$
$\text{These variations are not necessarily uniformly periodic in nature.} \\$
$\text{That is, they may or may not follow exactly similar patterns after equal}\\$
$\text{intervals of time. Generally one cyclic period ranges from 7 to 9 years}\\$
$\text{and there is no hard and fast rule in the fixation of years for a cyclic }\\$
$\text{period. For example, every business cycle has a Start- Boom- Depression-}\\$
$\text{Recover, maintenance during booms and depressions, changes in }\\$
$\text{government monetary policies, changes in interest rates.}\\$

$\text{8. Discuss about irregular variation}\\$
$\text{These variations do not have particular pattern and there is no regular}\\$
$\text{ period of time of their occurrences. These are accidently changes which are }\\$
$\text{purely random or unpredictable. Normally they are short-term variations, }\\$
$\text{but its occurrence sometimes has its effect so intense that they may give }\\$
$\text{rise to new cyclic or other movements of variations. For example }\\$
$\text{floods, wars, earthquakes, Tsunami, strikes, lockouts etc.} \\$

$\text{9. Define seasonal index.}\\$
$\text{A seasonal index is a statistical measure that shows the relative level of}\\$
$\text{ a variable (such as sales, demand, or production) in a specific season or } \\$
$\text{period, compared to the average level over all periods.In simple terms, it}\\$
$\text{ tells you how much a particular season differs from the average.}\\$
$\text{Seasonal Index}=\dfrac{\text{Seasonal Average}}{\text{Overall Average}} \times 100\\$

$\text{10. Explain the method of fitting a straight line.}\\$
$\text{(i) The straight line trend is represented by the equation Y = a + bX \; ....(1)} \\$
$\text{where Y is the actual value, X is time, a, b are constants.}\\$
$\text{(ii) The constants 'a' and 'b' are estimated by solving the two normal Equations} \\$
$\Sigma Y=n\;a+b\;\Sigma X \; ....(2)\\$
$\Sigma XY=a\; \Sigma X+b\;\Sigma X^{2} \; ....(3)\\$
$\text{Where 'n' = number of years given in the data.}\\$
$\text{(iii) By taking the mid-point of the time as the origin, we get} \; \Sigma X = 0\\$
$\text{(iv) When }\; \Sigma X = 0 ,\; \text{ the two normal equations reduces to} \\$
$\Sigma Y=n\;a+b(0)\; a=\frac{\Sigma Y}{n}=\bar{Y}\\$
$\Sigma XY=a(0)+b\; \Sigma X^{2}\; b=\frac{\Sigma XY}{\Sigma X^{2}} \\$
$\text{The constant 'a' gives the mean of Y and 'b' gives the rate of change (slope).}\\$
$\text{(v) By substituting the values of 'a' and 'b' in the trend equation (1),}\\$
$\text{ we get the Line of Best Fit.}\\$

$\text{ 11. State the two normal equations used in fitting a straight line.}\\$
$\text{ The two normal equations used in fitting a straight line are.}\\$
$\Sigma Y=n\;a+b\;\Sigma X \; ....(2)\\$
$\Sigma XY=a\; \Sigma X+b\;\Sigma X^{2} \; ....(3)\\$

$\text{12. State the different methods of measuring trend.} \\$
$\text{ Following are the methods by which we can measure the trend.} \\$
$\text{(i) Freehand or Graphic Method.}\\$
$\text{(ii) Method of Semi-Averages.}\\$
$\text{(iii) Method of Moving Averages.}\\$
$\text{(iv) Method of Least Squares.}\\$

$\text{13. Compute the average seasonal movement for the following series. }\\$

Year		Quarterly Production
	I	II	III	IV
2002	3.5	3.8	3.7	3.5
2003	3.6	4.2	3.4	4.1
2004	3.4	3.9	3.7	4.2
2005	4.2	4.5	3.8	4.4
2006	3.9	4.4	4.2	4.6

Year		Quarterly Production
	I	II	III	IV
2002	3.5	3.8	3.7	3.5
2003	3.6	4.2	3.4	4.1
2004	3.4	3.9	3.7	4.2
2005	4.2	4.5	3.8	4.4
2006	3.9	4.4	4.2	4.6
Quarterly Total	18.6	20.8	18.8	20.8
Quarterly Average	3.72	4.16	3.76	4.16

$\text{Grand average} = \dfrac{3.72+4.16+3.76+4.16}{4}\\$
$=\dfrac{15.8}{4}\\$
$=\text{Grand average} =3.95\\$
$\text{Seasonal Index =} \dfrac{\text{Quarterly average}}{\text{Grand average}}\times 100\\$
$\text{S. I for Quarter I} =\dfrac{3.72}{3.95}\times 100=94.18\\$
$\text{S. I for Quarter II} =\dfrac{4.16}{3.95}\times 100=105.32\\$
$\text{S. I for Quarter III} =\dfrac{3.76}{3.95}\times 100=95.19\\$
$\text{S. I for Quarter IV} =\dfrac{4.16}{3.95}\times 100=105.32\\$

$\text{14. The following figures relates to the profits of a commercial concern for 8 years.} \\$

Year	1986	1987	1988	1989	1990	1991	1992	1993
Profit (in Rs.)	15420	15470	15520	21020	26500	31950	35600	34900

$\text{ Find the trend of profits by the method of three yearly moving averages.} \\$

Year	Profit	3-yearly moving totals	3-yearly moving averages
1986	15420	-	-
1987	15470	46410	15470
1988	15520	52010	17336.67
1989	21020	63040	21013.33
1990	26500	79470	26490
1991	31950	94050	31350
1992	35600	102450	34150
1993	34900	-	-

$\text{15. Find the trend of production by the method of a five-yearly period} \\$
$\text{ of moving average for the following data:}\\$

Year	1979	1980	1981	1982	1983	1984
Production(in '000s)	126	123	117	128	125	124

Year	1985	1986	1987	1988	1989	1990
Production(in '000s)	130	114	122	129	118	123

Year	Production (in '000s)	5-yearly moving total	5-yearly moving averages
1979	126	-	-
1980	123	-	-
1981	117	619	123.8
1982	128	617	123.4
1983	125	624	124.8
1984	124	621	124.2
1985	130	615	123
1986	114	619	123.8
1987	122	613	122.6
1988	129	606	121.2
1989	118	-	-
1990	123	-	-

$\text{16. The following table gives the number of small-scale units registered}\\$
$\text{ with the Directorate of Industries between 1985 and 1991. Show the growth}\\$
$\text{ on a trend line by the free hand method.}\\$

Year	1985	1986	1987	1988	1989	1990	1991	1992
No. of units(in '000s)	10	22	36	62	55	40	34	50

$\text{17. The annual production of a commodity is given as follows :}\\$

Year	1995	1996	1997	1998	1999	2000	2001
Production(in tons)	155	162	171	182	158	180	178

$\text{Fit a straight line trend by the method of least squares.}\\$

Year (x)	Production (Y)	X=x-A (1998)	X²	XY	Trend Values
1995	155	-3	9	-465	159.56
1996	162	-2	4	-324	162.85
1997	171	-1	1	-171	166.14
1998	182	0	0	0	169.43
1999	158	1	1	158	172.72
2000	180	2	4	360	176.01
2001	178	3	9	534	179.3
n=7	ΣY=1186	ΣX=0	ΣX²=28	ΣXY=92

$a=\dfrac{\Sigma Y}{n} \\$
$a=\dfrac{1186}{7} \\$
$a=169.43 \\$
$b=\dfrac{\Sigma XY}{X^{2}} \\$
$b=\dfrac{92}{28} \\$
$b=3.29 \\$
$\text{Equation of trend line} =Y=a+bX \\$
$Y=169.43+3.29X \\$
$Y=169.43+3.29(x-1998) \\$
$When \;x=1995\; Y=169.43+3.29(1995-1998)\; Y=159.56 \\$
$When \;x=1996 \;Y=169.43+3.29(1996-1998)\; Y=162.85 \\$
$When \;x=1997 \;Y=169.43+3.29(1997-1998)\; Y=166.14 \\$
$When \;x=1998 \;Y=169.43+3.29(1998-1998)\; Y=169.43 \\$
$When \;x=1999 \;Y=169.43+3.29(1999-1998)\; Y=172.72 \\$
$When \;x=2000 \;Y=169.43+3.29(2000-1998)\; Y=176.01 \\$
$When \;x=2001 \;Y=169.43+3.29(2001-1998)\; Y=179.3 \\$

$\text{18. Determine the equation of a straight line which best fits the following data.}\\$

Year	2000	2001	2002	2003	2004
Sales(in '000s)	35	36	79	80	40

$\text{Compute the trend values for all years from 2000 to 2004}\\$

Year (x)	Sales (Y)	X=x-A (2002)	X²	XY	Trend Values
2000	35	-2	4	-70	43.2
2001	36	-1	1	-36	48.6
2002	79	0	0	0	54
2003	80	1	1	80	59.4
2004	40	2	4	80	64.8
n=5	ΣY=270	ΣX=0	ΣX²=10	ΣXY=54

$a=\dfrac{\Sigma Y}{n} \\$
$a=\dfrac{270}{5} \\$
$a=54 \\$
$b=\dfrac{\Sigma XY}{X^{2}} \\$
$b=\dfrac{54}{10} \\$
$b=5.4 \\$
$\text{Equation of trend line} =Y=a+bX \\$
$Y=54+5.4X \\$
$Y=54+5.4(x-2002) \\$
$When \;x=2000\; Y=54+5.4(2000-2002)\; Y=43.2 \\$
$When \;x=2001 \;Y=54+5.4(2001-2002)\; Y=48.6\\$
$When \;x=2002\;Y=54+5.4(2002-2002)\; Y=54 \\$
$When \;x=2003\;Y=54+5.4(2003-2002)\; Y=59.4 \\$
$When \;x=2004\;Y=54+5.4(2004-2002)\; Y=64.8 \\$

$\text{19. The sales of a commodity in tones varied from }\\$
$\text{ January 2010 to December 2010 as follows:} \\$

In Year 2010	Jan	Feb	Mar	Apr	May	June
Sales(in tons)	280	240	270	300	280	290
In Year 2010	July	Aug	Sept	Oct	Nov	Dec
Sales(in tons)	210	200	230	200	230	210

$\text{Fit a trend line by the method of semi average.}\\$

Month	Sale	Average
Jan	280
Feb	240
Mar	270	280+240+270+300+280+290/6=276.67
Apr	300
May	280
Jun	290
Jul	210
Aug	200
Sep	230
Oct	200	210+200+230+200+230+210/6=213.33
Nov	230
Dec	210

$\text{20. Use the method of monthly averages to find the monthly indices for the}\\$
$\text{following data of production of a commodity for the years 2002,2003 and 2004.}\\$

$\text{Grand average}=\dfrac{17.67+20.33+18+14.33+14+17+20.67+18+17.33+17+16.33+17.67}{12}\\$
$=\dfrac{208.33}{12}\\$
$\text{Grand average}=17.36\\$

$\text{Seasonal Index S.I}=\dfrac{\text{Monthly Average}}{\text{Grand Average}} \times 100\\$
$\text{S.I for Jan }=\dfrac{17.67}{17.36} \times 100=101.79\\$
$\text{S.I for Feb}=\dfrac{20.33}{17.36} \times 100=117.11\\$
$\text{S.I for Mar}=\dfrac{18}{17.36} \times 100=103.69\\$
$\text{S.I for Apr}=\dfrac{14.33}{17.36} \times 100=82.55\\$
$\text{S.I for May}=\dfrac{14}{17.36} \times 100=80.65\\$
$\text{S.I for Jun}=\dfrac{17}{17.36} \times 100=97.93\\$
$\text{S.I for Jul}=\dfrac{20.67}{17.36} \times 100=119.07\\$
$\text{S.I for Aug}=\dfrac{18}{17.36} \times 100=86.41\\$
$\text{S.I for Sept}=\dfrac{17.33}{17.36} \times 100=99.83\\$
$\text{S.I for Oct}=\dfrac{17}{17.36} \times 100=97.93\\$
$\text{S.I for Nov}=\dfrac{16.33}{17.36} \times 100=94.07\\$
$\text{S.I for Dec}=\dfrac{17.67}{17.36} \times 100=101.79\\$

$\text{21. Calculate the seasonal indices from the following data using the }\\$
$\text{average from the following data using the average method:}\\$

	Quarter I	Quarter II	Quarter III	Quarter IV
2008	72	68	62	76
2009	78	74	78	72
2010	74	70	72	76
2011	76	74	74	72
2012	72	72	76	68

	Quarter I	Quarter II	Quarter III	Quarter IV
2008	72	68	62	76
2009	78	74	78	72
2010	74	70	72	76
2011	76	74	74	72
2012	72	72	76	68
Quarterly Total	372	358	362	364
Quarterly Average	74.4	71.6	72.4	72.8

$\text{Grand average}=\dfrac{74.4+71.6+72.4+72.8}{4}\\$
$=\dfrac{291.2}{4}\\$
$\text{Grand average}=72.8\\$

$\text{Seasonal Index S.I}=\dfrac{\text{Quarterly Average}}{\text{Grand Average}} \times 100\\$
$\text{S.I for Quarter I}=\dfrac{74.4}{72.8} \times 100=102.20\\$
$\text{S.I for Quarter II}=\dfrac{71.6}{72.8} \times 100=98.35\\$
$\text{S.I for Quarter III}=\dfrac{72.4}{72.8} \times 100=99.45\\$
$\text{S.I for Quarter IV}=\dfrac{72.8}{72.8} \times 100=100\\$

$\text{22. The following table shows the number of salesmen working for a certain concern:}\\$

Year	1992	1993	1994	1995	1996
No. of salesmen	46	48	42	56	52

$\text{ Use the method of least squares to fit a straight line }\\$
$\text{and estimate the number of salesmen in 1997.}\\$

Year (x)	Sales (Y)	X=x-A (2002)	X²	XY	Trend Values
1992	46	-2	4	-92	44.8
1993	48	-1	1	-48	46.8
1994	42	0	0	0	48.8
1995	56	1	1	56	50.8
1996	52	2	4	104	52.8
n=5	ΣY=244	ΣX=0	ΣX²=10	ΣXY=20

$a=\dfrac{\Sigma Y}{n} \\$
$a=\dfrac{244}{5} \\$
$a=48.8\\$
$b=\dfrac{\Sigma XY}{X^{2}} \\$
$b=\dfrac{20}{10} \\$
$b=2 \\$
$\text{Equation of trend line} =Y=a+bX \\$
$Y=48.8+2X \\$
$Y=54+5.4(x-1994) \\$
$When \;x=1992\; Y=48.8+2(1992-1994)\; Y=44.8\\$
$When \;x=1993 \;Y=48.8+2(1993-1994)\; Y=46.8\\$
$When \;x=1994\;Y=48.8+2(1994-1994)\; Y=48.8\\$
$When \;x=1995\;Y=48.8+2(1995-1994)\; Y=50.8 \\$
$When \;x=1996\;Y=48.8+2(1996-1994)\; Y=52.8 \\$

Applied Statistics Exercise 9.1

Applied Statistics Exercise 9.1

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Applied Statistics Exercise 9.1

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