Maths

University of Development Alternative (UODA)

Correlation

1.A computer whole calculating the correlation coefficient between the variables X and Y obtain the following result:N=30, X=120, Y=90, XY=335, X2=600, Y2=250, It was, however, later discovered at the time of checking that it had copied down two pairs of observations as xy

810

127

While the correct values were xy

812

108

Obtain the correct value of the correlation coefficient between the variables X and Y.2. An office contains 9 officers. The long serving officers feel that they should have a seniority increment based on length service built into their salary structure. An assessment of their efficiency by their department manager and personnel department produces a ranking of efficiency. This is shown below together with a raking of their length of services. Do the data support the officers, claim for seniority increment?Ranking according

to length of services:123456789 Ranking according

to efficiency :235191011873. A company gives on-the-job training to its salesmen which are subject to a test. The company terminates the service of its salesmen who do not do well in the test.

The following data give the test scores and sales made by nine salesmen during the last one year:

Test scores

:141924212622152019

Sale (Tk. 000):313648375045334139 Compute the correlation of coefficient between test scores and sales. Does it indicate that termination of the services of salesmen with low test scores is justified?4.Following figures give the rainfall in inches for the year and the production in 00s of kfs. for the Rabi crop and Kharip crop. Calculate the Karl Pearsons coefficient of correlation, between rainfall and total crop:

Rain fall

:20222426283032

Rabi Production:15182032403940

Kharif Production:15172018202115 5. Calculate the correlation coefficient between age and playing habits of the following students and comment on it. Also calculate the probable error:

7

Age

:1213141516

No. of students:250200120150100

Regular players:200150489050

6.The city corporation in Bangladesh is considering increasing the number of police in an effort to reduce crimes. Before taking a final decision, the corporations council has asked the chief of the police commissioner to survey the entire city corporation to determine the relationship between the police and the number of crimes reported. The chief commissioner gathered the following information: 5

City

Police

No of Crimes

Dhaka

15

17

Chittagong

17

13

Rajshahi

25

10

Khulna

27

7

Barishal

17

5

Calculate the correlation coefficient between the numbers of crimes and the numbers of police and interpret.

7. The scores of students in an examination in Math. and Stat. is given below:Students No:123456789

Marks in Math:704858555450605240

Marks in Stat.:624753605568514843

Calculate rank correlation coefficient between Mathematics and Statistics and compare the two values.

Example-01: The supply and price of a market on one month are given below. Calculate their coefficient of correlation.Supply (in ton)80828691838589

Price(per 10 kg)146140130117133127115

Example -02: The following table shows the marks of Statistics and mathematics of 10 Students of BBA Department of UODA. Calculate the Rank correlation and comment on your result.Statistics92898786837771635350

Mathematics86839177688552823757

Example -03: Two teaching methods A and B are applied on 11 students and the required numbers are given below. Find rank correlation coefficient.

Students1234567891011

Marks of A2429191430192731202819

Marks of B3735162623271920161121

11. The number of Statistics and Mathematics of 8 students are given below. Calculate their correlation

coefficient, Rank correlation and comment on your result.

Statistics8030902550708288

Mathematics7040505355657560

12. Following information shows the use of fertilizer and production of rice of eight fields.

Calculate coefficient of correlation and comment on your result.

Fertilizer used (in kg)1015202530354045

Rice produced (in kg)200250300340370390400405

13. Following information shows the use of cigarette and probabilities of cancer of seven patients.

Calculate coefficient of correlation and comment on your result.

Number of cigarette10131725323540

Probabilities of cancer.20243038444955

Regression Analysis1. The following data give the ages and blood pressure of eight women.

Age(x)

: 56 58 64 67 79 Blood Pressure (y): 87 90 98108125

i. Find the correlation coefficient between x and y.

ii. Determine the least square regression equation of y on x

iii. Estimate the blood pressure of a woman whose age is 45 years.2. After investigation it has been found the demand for automobiles in a city depends mainly, if not entirely, upon the number of families of residing in that city. Below are given figures for the sales of automobiles in the five cities for the year 2010 and the number of families residing in those cities:CityNo. of families in lakhs(x)Sales of automobiles in 000 (y)

A7025

B7528

C8030

D6022

E9035

Fit a linear regression equation of y on x by the least square method and estimate the sales for the year 2013 for city A which is estimated to have 100 lakh families assuming that the same relationship hold true.3. Find the most likely production corresponding to a rainfall of 40 inches from the following data:

RainfallProduction

Average 30 inches50 quintals

S.D05 inches10 quintals

Correlation of Coefficient0.8

4. The General sales manager of Kiran Enterprises-an enterprise dealing in the sales of ready-made mens wears is toying with the idea of increasing his sales toTk.80000. On checking the records of ales during the last year 10 years, it was found that the annual sale proceeds and advertisement expenditure were highly correlated to the extent of 0.8. It was further noted that the annual average sale has been Tk.45000 and annual average advertisement expenditure Tk.30000, with a variance of Tk. 1600 and Tk.626 in advertisement expenditure respectively.In view of the above, how much expenditure of advertisement you would suggest the General Sales Manager of the enterprise to incur to meet his target of sales.5. In a partially destroyed laboratory record of an analysis of correlation data, the following results only are legible:Variance of x=9Regression equation=8X-10Y+66=040X-18Y=214Find on the basis of the above information:

1. The mean values of X and Y.

2. Coefficient of correlation between X and Y.

3. Standard deviation of Y.6. A financial analyst has gathered the following data about the relationship between income and investment in respect of 5 randomly selected families:Income:81292437

Percent:3625331519

Invested in securities

Develop an estimating equation that best describes these data. Find the coefficient of determination and interpret it.

Calculate the standard error of estimate for this relationship.

Find an approximate 90 percent confidence interval for the percentage of income invested in securities by a family earning Tk.2500 annually. 7. The following information is collected from a super shop regarding their sales and advertising expenditures in one year.

Ad. Expenditure (tk. Thousand)Sales (tk. Thousand)

Average50100

Standard Deviation3625

Coefficient of Correlation0.85

i. Calculate two regression equations.ii. Estimate the approximate sales for a proposed advertisement expenditure of tk. 120.iii. What should be the advertisement budget if the company wants to achieve a sales target of tk. 400 thousand?

8. Catalogues listing text books were examined to discover the relationship between the cost of a book and the number of pages it contains. The perusal gives the following data for seven books:

Pages

:40455055606570

Prices (Tk.):10141820222530

What increase would you expect for a book if it is decided to increase the number of pages of the book by 50? 9. The following information is collected from a super shop regarding their sales and advertisement expenditure of one year (300 working days are considered one year).

Sales (TK. Thousand)Advertisement Expenditure (TK. Thousand)

Arithmetic Mean985



i) Calculate total sales and total advertisement expenditure of one year.

ii) Find two regression equations

iii) Estimate the approximate sales for a proposed advertisement expenditure of TK. 7 thousand.

iv) What should be the advertisement budget if the company wants to achieve a sales target of TK. 150 thousand?

10. Following information shows the use of fertilizer and production of rice of eight fields.a) Estimate two regression lines and comment of your results.b) What will be the estimated rice production for 60 of fertilizer?c) If one wants to get 600 of rice production calculate the approximate fertilizer to use.Fertilizer used (in kg)1015202530354045


11. You are given the following data about rain fall and production of rice of 35 fields.Rain fall (cm)Production (kg)

Arithmetic Mean26.75084

Standard Deviation4.636.8

Coefficient of CorrelationRxy = 0.52

Calculate two regression lines. Estimate the production of rice if rain fall is 22cm and estimate the rainfall if the production is 600 kg.

Example 02: The following information is collected from a super shop regarding their sales and advertisement expenditure of one year (300 working days are considered one year).

Sales (Tk. thousand)Advertisement Expenditure (Tk. Thousand)

Arithmetic Mean985



i) Calculate total sales and total advertisement expenditure of one year.

ii) Find two regression equations

iii) Estimate the approximate sales for a proposed advertisement expenditure of Tk. 7 thousand.

iv) What should be the advertisement budget if the company wants to achieve a sales target of Tk. 150 thousand?

v) 8. Following information shows the use of fertilizer and production of rice of eight fields.

vi) a) Estimate two regression lines and comment of your results.

vii) b) What will be the estimated rice production for 60 of fertilizer?

viii) c) If one wants to get 600 of rice production calculate the approximate fertilizer to use.

Fertilizer used (in kg)1015202530354045


ix) 9. You are given the following data about the sales and Advertisement expenditure of one year (300

x) working days) of a firm.

Sales (Tk. Crores)Advertisement Expenditure (Tk. Crores)

Arithmetic Mean5010


Number Days300


xi) a) Calculate Total Sales and total Advertisement expenditure of the year.

xii) b) Calculate two regression equations

xiii) c) Calculate Coefficient of Regression of sales on advertisement expenditure and comment.

xiv) d) Estimate the approximate sales for a proposed advertisement expenditure of Tk. 13.5 crore.

xv) e) What should be the advertisement budget if the company wants to achieve a sales target of Tk.

xvi) 70 crore?

xvii) 10. You are given the following data about rain fall and production of rice of 35 fields.

Rain fall (cm)Production (kg)

Arithmetic mean26.75084

Standard deviation4.636.8

Coefficient of correlation

xviii) Calculate two regression lines. Estimate the production of rice if rain fall is 22cm and estimate the

xix) rainfall if the production is 600 kg.

Time Series Analysis1. Below are given the figures of production (in thousand quintals) of a sugar factoryYear:1990199119921993199419951996

Production: 80 90 92 83 94 99 92

( in ooo qtl)

Fit a straight line tend to these figure.

Plot these figure on graph and show the trend line.

Estimate the likely sales of the company during 2000.

Eliminate the trend. What components of the Time Series are thus left over?

What is the monthly increase in the production of sugar? 2.Below are given the figures of production (in million tones ) of a cement factory

Year:1990199219931994199519961999Production: 80 90 92 83 94 99 92

( in ooo qtl)

Fit a straight line tend to these figure.

Plot these figure on graph and show the trend line.

Estimate the likely sales of the company during 2000.

Eliminate the trend. What components of the Time Series are thus left over?

What is the monthly increase in the production of sugar?

3.The working capital requirements of the XYZ Ltd. Have been subject to seasonal fluctuations. At the same time, a steady secular advance can be noted. In oder to evaluate comprehensively future working cpital needs, the treasure calculated a straight line trend and the seasonal indices. The trend equation is Y=10000=500X, where X represents a period of 1 month and has a value of 0 in 2000. The seasonal indices are as follows:

Jan.80

July125

Feb.95

Aug.99Mar.90

Sep.90

Apl.100

Oct.102

May.116

Nov.105

June.120

Dec.87 Prepare a schedule of estimated working capital requirements for 2001. What factor could cause these estimates to be incorrect?

What might be done to compensate for inaccuracies as they become apparent?

Would you as a banker have any interest in estimates of this type?

4.The sale of a commodity in ton varied from January 2005 to December 2005 in the following manner:

MonthSales( inton)

Month

Sales( inton) Jan. 280

July

225

Feb. 300

Aug.

210

Mar. 290

Sep.

290

Apl. 250

Oct.

210May. 216

Nov.

205

June. 220

Dec.

280 Fit a trend by the method of semi-average. Fit a trend by the method of 3 or 4 monthly moving average.

5.Compute a nonlinear trend of the form Y=a+bx+cx2 for the data showing the production of wheat ( in thousand tones) during the years 1992 to 2000.Year

:199219931994199519961997199819992000

Production: 9 10 12 15 13 10 8 16 15 of wheat(in ton)

Probability

1.A sample survey conducted in four cities pertained to preference for brand A soap. The response is shown below:

Dhaka

khulna

RajshahiNoakhali

Yes

: 35

40

55

60

No

: 20

30

25

45

No response: 04

06

02

05

What is the probability that a customer selected at random 6

Does not preferred brand A.

Preferred brand A and was from Dhaka.

Given that he preferred brand A, what is the probability that he was from khulna?

Preferred brand A given that he was from Dhaka.

2. Two balance dice, one is black and the other is red, are thrown and the number of dots on their upper faces are noted. Let b be the outcomes of the black die and r be the outcomes of the red die. Find following probabilities: 7

P(b=2r), P(b=odd number, r=4), P(b+r=14), P(b+r=5, br=6)

3. From a lot of 20 items containing 3 defective, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Answer the probability P(0X

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