OP Malhotra Class 11 Maths Solutions Chapter 31 Moving Average Ex 31

Effective Class 11 ISC Maths OP Malhotra Solutions Chapter 31 Moving Average Ex 31 can help bridge the gap between theory and application.

S Chand Class 11 ICSE Maths Solutions Chapter 31 Moving Average Ex 31

Question 1.
The table shows the number of students in a school getting at least a grade C in mathematics for the years 1994 to 2001.
(i) Represent this data as a time series.

1994	1995	1996	1997	1998	1999	2002	2001
97	118	115	117	121	125	111	125

(ii) Calculate the 3-point moving average and plot it on the same graph.
(iii) Are the school’s maths results improving?
(iv) Explain why this is not a good way to work out whether the school’s results are improving.
Solution:
(i) The table of values is given as under :


(iii) The set of data is smoothed out and is increasing.
(iv) The number is increasing but might be because the school is getting bigger.

Question 2.
The profits of a soft drink firm in thousands of litres during each month of a year were :

Calculate 3-monthly moving averages and illustrate graphically. (ISC 1992 type)
Solution:
Calculation of 3-monthly moving arrange is given as :

Month	Profit	3-monthly moving total	3-monthly moving average
Jan.	1.2	–	–
Feb.	0.8	3.4	1.13
Mar.	1.4	3.8	1.267
Apr.	1.6	5.0	1.67
May	2.0	5.6	1.87
June	2.0	7.6	2.53
July	3.6	10.4	3.47
Aug.	4.8	11.8	3.93
Sep.	3.4	10.0	3.33
Oct.	1.8	5.9	1.97
Nov.	0.7	3.7	1.23
Dec.	1.2	–	–

Question 3.
The number of traffic offences committed in a certain city over a period of 3 years is given in the following table :

	Jan-March	April-June	July-Sept.	Oct-Dec.
1968	74	56	48	69
1969	83	52	49	81
1970	94	60	48	79

Calculate 4-quarterly moving averages and illustrate these and original figures on one graph using the same axis for both. Comment briefly on a local politician’s claim that traffic offences were on the increase.
Solution:

Thus the local politician’s claim that traffic offences were on the increase be true, it is cleared from 4-quarterly moving average centred column.

Question 4.
Find the 4-quarterly moving averages in the following table which gives the quarterly index numbers of coal production (for the years 1936-1938). Also plot on the same graph the quarterly index numbers as well as the 4-quarterly moving average. Comment on the nature of the general trend.

Year	Quarters
	1	2	3	4
1936	93.3	81.7	81.5	89.1
1937	93.8	92.3	86.5	93.7
1938	97.6	82.3	79.0	89.3

Solution:

Question 5.
The annual incomes of a firm were recorded every quarter for 4 years. The results are shown in this table.

	1999	2000	2001	2002
1st quarter	₹ 18,00,000	₹ 20,00,000	₹ 21,00,000	₹ 22,50,000
2nd quarter	₹ 14,50,000	₹ 17,80,000	₹ 19,50,000	₹ 21,00,000
3rd quarter	₹ 13,50,000	₹ 15,00,000	₹ 18,00,000	₹ 19,80,000
4th quarter	₹ 19,00,000	₹ 18,30,000	₹ 19,20,000	₹ 20,50,000

(i) Work out the 4-point moving average for the data.
(ii) Plot the original data and the moving average on the same graph.
(iii) Comment on how the firm’s incomes have changed over the 4-years.
Solution:


The data has been smoothed out so there is a steady increase in Income.

Question 6.
The following table shows the daily sales of milk at a local corner shop for a month.

Sun	Mon	Tue	Wed	Thu	Fri	Sat
12	8	6	9	4	11	15
11	7	7	6	3	15	14
14	9	7	7	5	12	15
11	12	8	7	4	14	19

Make a table showing the moving average using a 7-day span, and draw a graph to show the trend of milk sales over the month.
Solution:

Question 7.
The following table gives the monthly expenditure on a motor car for a period of two years.

Calculate 12-month!y moving average for the two years and display them and the original table on the same graph.
Solution:


Question 8.
A new film was shown at a theatre and ran for six w eeks. The attendances are shown in the table.

	Mon	Tue	Wed	Thu	Fri	Sat
First week	243	268	407	384	348	489
Second week	445	501	623	621	527	684
Third week	602	625	800	763	728	800
Fourth week	800	800	800	800	800	800
Fifth week	721	785	800	800	800	800
Sixth week	647	664	683	642	608	726

(i) Plot a line graph from the above data.
(ii) Calculate the 6-day moving average for the data and plot this on the same graph.
(iii) Comment on the weekly attendance.
Solution:

Question 9.
The table below given details of the electricity generated in million kilowatt hoars for public supply in each quarter of the years 1952 to 1955.

Year	Quarter
	1	2	3	4
1952	8.9	7.1	6.7	9.3
1953	10.1	7.5	7.1	10.5
1954	11.7	7.5	8.3	10.9
1955	12.5	8.3	9.5	11.7

Draw a graph illustrating these figures.
Calculating a set of moving averages using the most suitable number of observations; give reasons of your choice. On the same diagram as before draw a graph showing the moving averages.
Solution:
We shall use 4-monthly moving average as it null eliminate the 12 monthly cycle and leave the general trend of the data.

Question 10.
The number of letters, in hundreds, posted in a certain city on each day of a fornight was as follows:

	Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
First week	35	70	36	59	62	60	71
Second week	39	72	38	56	63	71	75

Calculate the 7-day moving averages and display these and the original figures graphically on the same diagram, using the same scale and axes. What is the general trend ?
Solution:

Week days	Week days	No. of letters	7-day moving total	7-day moving average
First	Sun	35
	Mon	70
	Tue	36
	Wed	59	393	56.14
	Thu	62	397	56.71
	Fri	60	399	57
	Sat	71	401	57.28
Second	Sun	39	398	56.85
	Mon	72	399	57
	Tue	38	410	58.57
	Wed	56	414	59.14
	Thu	63
	Fri	71
	Sat	75

From graph and table, we observe that the general trend is that the no. of letters posted goes on increasing every day week after week.

Question 11.
In an influenza epidemic the numbers of cases diagnosed were :

Date(Marks)	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Numbers	2	0	5	12	20	27	46	30	31	18	11	5	0	1

On what days do the mode and upper and lower quartiles occur ?
Calculate 3-day moving averages and display them and the original figures on the same graph.
Solution:
Calculation of 3 yearly moving average

Question 12.

Year	Date rate per thousand Quarter ended
	March	June	September	December
1953	13.9	10.3	8.1	10.6
1954	13.8	9.8	7.8	10.8
1955	14.2	10.1	7.8	10.0

Plot these figures on a graph.
Calculate the 4-quarterly moving averages and plot on the same graph.
Solution:

Question 13.
Registered unemployed (hundreds)

	1957	1958
January	638	596
February	602	548
March	509	491
April	462	462
May	359	365
June	295	325
July	290	308
August	322	328
September	377	377
October	392	380
November	480	474
December	542	536
Average for year	439	432.5

Plot these monthly figures on a graph. Calculate the 12-monthly moving averages and plot these on the same graph.
Solution:

Question 14.
A Ballet Company gave a 6-weeks’ season at a large hall capable of seating 4000 people and the attendances in hundreds, at the evening performances, are recorded in the following table.
Attendance, in hundreds, to nearest hundred

	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
First week	12	13	20	19	17	24
Second week	22	25	31	31	26	34
Third week	30	31	40	38	36	40
Fourth week	40	40	40	40	40	40
Fifth week	38	39	40	40	40	40
Sixth week	32	33	34	32	30	36

Plot a graph of the above time-series and include on the same diagram the graph of 6-day moving averages.
Comment on the weekly cycle on attendances and state, with reasons, if you think, an extension of the season of eight weeks, would have been justified.
Solution:

Question 15.
Production of passenger cars, U.S.A. (tens of thousands)

Year	Quarter
	I	II	III
1927	26	36	24
1928	29	36	36
1929	40	52	43

Calculate the 4-quarterly moving averages and then draw the graphs of the given series and the moving averages. Briefly comment on the general trend.
Solution:

From table and graph it is observed that, the demand of cars was increasing year after year.

Question 16.
The aggregate number, in millions, of working days lost in strikes during each year of the period 1950-60 was

1950	‘51	‘52	‘53	‘54	‘55	‘56	‘57	‘58	‘59	‘60
1.4	1.7	1.8	2.2	2.5	3.8	2.1	8.4	3.5	5.3	3.0

Draw a graph to represent this information. Calculate the 3-yearly moving averages and draw the 3-yearly moving averages graph, using the same axes and scales. What is the main purpose in drawing moving average graph ? Comment on whether the purpose is achieved in this case.
Solution:

Years	Working day lost	3-yearly moving total	3-yearly moving average
1950	1.4
1951	1.7	4.9	1.63
1952	1.8	5.7	1.9
1953	2.2	6.5	2.17
1954	2.5	8.5	2.83
1955	3.8	8.4	2.8
1956	2.1	14.3	4.77
1957	8.4	14.0	4.67
1958	3.5	17.2	5.73
1959	5.3	11.8	3.93
1960	3.0

The main purpose in drawing moving average graph is to find the general trend. The purpose is achieved in this case, because the graph shows that the number of working lost during strike is goes on increasing.

Question 17.
The profits of a soft drink firm in thousands of rupees during each month of a year were

Jan.	Feb.	Mar.	Apr.	May	June	July	Aug.	Sept.	Oct.	Nov.	Dec.
1.2	0.8	1.4	1.6	2.0	2.4	3.6	4.8	3.4	1.8	0.8	1.2

Plot these on a graph.
Calculate 4-monthly moving averages and plot these on the same graph. Comment on the general trend.

The general comment on trend is that the profits go on increasing from Jan. to August and start decreasing from Sept, to December.

Question 18.
Calculate,5-yearIy moving averages for the following data of the commercial and industrial failures in a country from 1982 to 1997.

Year	1982	1983	1984	1985	1988	1989	1990	1991	1992	1993	1996	1997
No. of families	23	26	28	32	12	10	9	13	11	14	3	1

Display the actual and tend values on the same graph using the same axes for both.
Solution:

Year	No. of failures	5-yearly moving total	5-yearly moving average
1982	23
1983	26
1984	28	121	24.2
1985	32	108	21.6
1988	12	91	18.2
1989	10	76	15.2
1990	9	55	11
1991	13	57	11.4
1992	11	50	10
1993	14	42	8.4
1996	3
1997	1

Question 19.
The table given below shows the daily attendance in thousands at a certain exhibition over a period of two weeks :

Week 1	52	48	64	68	52	70	72
Week 2	55	47	61	65	58	75	81

Calculate 7-day moving averages and illustrate these and original information on the same graph using the same scales.
Solution:
Calculation of 7-yearly moving average

Question 20.
The profit of a soft-drink firm (in thousands of rupees) during each month of the year is as given below :

Jan.	Feb.	Mar.	Apr.	May	June	July	Aug.	Sept.	Oct.	Nov.	Dec.
3.6	4.3	4.3	3.4	4.4	5.4	3.4	2.4	3.4	1.8	0.8	1.2

Calculate the 4-monthly moving averages and plot these and the original data on a graph sheet.
Solution:
Calculation of 4 yearly moving average