PROBLEM SHEET

1. A sales manager has experienced with four different sales approaches and has a large group of salesmen applying each of four approaches (A, B, C and D). He samples at random from each group, thus obtaining the data shown below in the table. Analyse this data using analysis of variance (0.05 level) in testing whether any real difference exists between the four sales approaches as they relate to the successful completion of sales.

Rupees of Sale Resulting from Application of Four Sales Approaches

Sales ApproachA B C D

121116141013

14151312

10911879

1712141310

Total 76 54 54 66

2. A marketing researcher wishes to test whether any significant difference exists between the performances of three retail stores in their sales of a particular product. Test whether the average of the retail store sales are equal at the 0.05 level. Tubular values represent in Rs. lakhs.

Retail StoresI II III

121461213

16151719

13121091413

3. Four different give-away premiums are being tried in a test area to determine if any significant difference exists in their ability to attract buying customers. Sales are recorded for the stores using each of the four premiums. Test at the 0.01 level. Tubular values represent sales in Rs. lakhs.

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PremiumI II III IV

3542632113

4860333940

43495550

504336428070

4. An investor has some money which he wants to invest in shares. Obviously, he would like to invest in those shares promise higher returns. Also, he doesn’t want to invest all his money in one; rather he wants to distribute it in three shares. He has collected the following information on five different shares in ten randomly selected time periods. Can you help the investor in deciding which three shares to invest his money in? (Note: The last two rows give the column sum and column SS respectively).

RATE OF RETURN INDCM JK LCP ABC XYZ28.32 40.65 33.98 33.75 27.7322.71 30.62 43.61 24.64 26.7314.41 32.98 36.77 19.77 32.5922.82 24.80 31.56 30.79 24.3526.92 32.17 37.00 33.54 30.1531.46 25.23 40.00 24.44 31.1026.80 21.83 27.60 25.75 33.6816.32 29.64 34.60 18.51 31.8426.68 32.05 48.88 30.14 25.0121.77 35.54 34.07 24.97 29.31

237.71 305.51 367.98 266.30 291.475878.6 9609.6 13876.0 7344.9 8585.0

5. Six competing brands of two-wheelers are compared on their mileage on city roads. Ten users of each two-wheeler are selected at random and data on mileage are collected from them. After scrutiny, 2 observations for brand 2, 1 observation of brand 3 and 3 observations of brand 5 are discarded from consideration. The sample means and standard deviations of mileage for the six different brands of two-wheelers are then calculated which are given below. Test at a 5% level of significance that there is no significant difference in the average mileage of the six brands of two-wheelers under consideration.

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Two-wheeler Brand 1 2 3 4 5 6Valid Sample Size 10 8 9 10 7 10Mean Mileage 41.87 40.10 44.56 50.10 40.49 47.30SD of Mileage 8.67 8.20 5.79 10.24 10.51 12.17

TWO-WAY ANALYSIS OF VARIANCE

6. A researcher wishes to test four sale techniques to determine whether a significant difference exists between them. He reasons that the size of store will make a substantial contribution to total variability and decides to stratify his stores according to size, the strata or blocks being large, medium, and small stores. He then randomly selects four stores from each block and randomly assigns a different treatment (sales technique) to each of the four stores in an individual block. Such an approach assures him that every sales technique is tried in each size of store. The resultant sale are shown in the table below:-

Sales Resulting from Test of Four Sales Techniques Randomized BlockDesign-one Observation per Cell

Sales technique

Size of store A B C D Total

I LargeII MediumIII Small

201612

251814

232016

221915

907357

Total 48 57 59 56 220

7. The Sales and Promotions Manager of a company which sells a well-known drink wishes to know whether any difference exists between promotion campaigns on college campus. He is also interested in knowing whether differences exist in type of college (day, residential, and semi-residential). The Sale Manager suspects that some interaction may be occurring between the two classifications. Make the necessary tests for the following data at the 0.01 level.

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Promotional technique

College type A B C DDay 20

2518

222419

171815

101514

Residential 121516

282725

232521

171819

Semi-residential 171926

162120

303226

232427

8. A product was test-marketed to identify the market segment. The entire market was divided into 5 zones and in each zone, six income categories were considered. The sales data is tabulated below. The manufacturer of the product does not believe that, in the target population, incomes vary significantly across the zones. Would you conclude that sales are significantly different in all the zones and in all the income categories? (Overall s.d. = 6.86).

Inc.Cat.

Zone Sales IncCat.

Zone Sales IncCat.

Zone Sales

1 1 13 3 1 21 5 1 401 2 23 3 2 19 5 2 171 3 27 3 3 17 5 3 261 4 14 3 4 21 5 4 281 5 28 3 5 19 5 5 202 1 24 4 1 28 6 1 172 2 22 4 2 21 6 2 182 3 24 4 3 28 6 3 362 4 16 4 4 17 6 4 332 5 38 4 5 25 6 5 20

9. A new HRD Manager of a company is very enthusiastic about his job. He has worked out an all round development plant for the workers. But while discussing the plan with the workers, he finds that most of them are very reluctant to any training as they strongly believe that a worker’s salary is dependent only on the number of years one has spent in the company rather than how skilled one is to do specific jobs. In order to investigate into the workers’ complaints, he HRD manager selects exactly one employee from each skill and experience combination category and obtains the following

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information, where mean salary is the mean salary in nearest thousand rupees. The overall standard deviation of salary for the entire sample is 6.36. On the basis of the data, would you conclude that the workers’ complaints were right?

Level of Skill Mean salary Experience (in year)

Mean Salary

Unskilled 29.50 < 3 22Semi-skilled 25.75 3-5 26Skilled 26.75 5-8 32High skilled 31.75 > 8 34Very High skilled

28.75

10. RELIABLE tyre dealer supplying three brands of tyres (T1, T2, T3) to

automobile manufacturing company wishes to identify suitability of each brand of its tyres for the four brands of car (C1, C2, C3, C4). Practically, the dealer to determine whether one particular brand of tyre more mileage than others for a particular brand of car. For this purpose, dealer collects data on mileage (in ‘000 kilometers) for each brand of tyre on each of the four brands of cars. The same are presented below:-

TYRE

BRAND

T1

CAR BRAND C1 C2 C3 C4

30 34 38 36 40 40 39 38 42 42 40 39 33 45 42 40

T2

35 36 40 41 39 38 43 39 41 42 41 40 39 43 32 42

T3

34 44 30 45 38 45 28 43 39 37 32 42 35 38 29 46

Analyze the date and draw relevant inferences at œ = 0.05

[Given that Pr{F(2,36)>3.25945}=Pr{F(3,36)>2,86627} = Pr{F(6,36>2.36375}=0.05]

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