SEM-1_MCom_Quantative Techniques for Managers_Unit12_ANOVA

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2 Quantative Techniques for Managers www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

3 Unit 12 – ANOVA ▪ ANOVA- Concept &Related Terms ▪ Solved example ▪ Extra – Steps www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

4 ANOVA- Concept & Related Terms www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

Analysis of variance: 5 ▸ A statistical procedure for determining whether the means of several different populations are equal. www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL 5

Concepts: 6 • A sampling plan or experimental design is the way that a sample is selected from the population under study and determines the amount of information in the sample. • An experimental unit is the object on which a measurement or measurements is taken. Any experimental conditions imposed on an experimental unit provides effect on the response. www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL 6

Contd… 7 • A factor or criterion is an independent variable whose values are controlled and varied by the researcher. • A level is the intensity setting of a factor. • A treatment or population is a specific combination of factor levels. • The response is the dependent variable being measured by the researcher. www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL 7

For example, 8 1. A tyre manufacturing company plans to conduct a tyre-quality study in which quality is the independent variable called factor or criterion and 2. the treatment levels or classifications are low, medium and high quality. www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL 8

9 ▸ The dependent (or response) variable might be the number of kilometers driven before the tyre is rejected for use. ▸ A study of daily sales volumes may be taken by using a completely randomized design with demographic setting as the independent variable. ▸ A treatment levels or classifications would be inner-city stores, stores in metro-cities, stores in state capitals, stores in small towns, etc. The dependent variable would be sales in rupees. www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL 9

More Concepts 10 ▸ Factor: Another word for independent variable of interest that is controlled in the analysis of variance. ▸ Factor level: A value at which the factor is controlled. www.cuidol.in Unit-1(MAP-607) 10 All right are reserved with CU-IDOL

Assumption for ANOVA 11 www.cuidol.in Unit-1(MAP-607) 11 All right are reserved with CU-IDOL

One-way analysis of variance: 12 ▸ Analysis of variance in which only one criterion (variable) is used to analyze the difference between more than two population means. www.cuidol.in Unit-1(MAP-607) 12 All right are reserved with CU-IDOL

▸ Example 1: Three brands A, B, and C of tyres were tested for 13 durability. A sample of four tyres of each brand is subjected to the same test and the number of kilometres until wear out was noted for each brand of tyres. The data in thousand kilometres is given in www.cuidol.in Unit-1(MAP-607) 13 All right are reserved with CU-IDOL

14 ▸ Since the same number of observations is obtained from each brand of tyres (population), therefore the number of observations in the table is n = rk = 3×4 = 12. ▸ www.cuidol.in Unit-1(MAP-607) 14 All right are reserved with CU-IDOL

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17 ▸ Example 12.1: To test the significance of variation in the retail prices of a commodity in three principal cities, Mumbai, Kolkata, and Delhi, four shops were chosen at random in each city and the prices who lack confidence in their mathematical ability observed in rupees were as follows: www.cuidol.in Unit-1(MAP-607) 17 All right are reserved with CU-IDOL

18 ▸ Do the data indicate that the price in the three cities are significantly different? www.cuidol.in Unit-1(MAP-607) 18 All right are reserved with CU-IDOL

19 ▸ Solution: Let us take the null hypothesis that there is no significant difference in the prices of a commodity in the three cities. Calculations for analysis of variance are as under. www.cuidol.in Unit-1(MAP-607) 19 All right are reserved with CU-IDOL

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21 ▸ There are r = 3 treatments (samples) with n1 = 4, n2 = 4, n3 = 4, and n = 12. www.cuidol.in Unit-1(MAP-607) 21 All right are reserved with CU-IDOL

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24 ▸ ANOVA Table: www.cuidol.in Unit-1(MAP-607) 24 All right are reserved with CU-IDOL

25 ▸ The table value of F ifsor4.d2f61 .=S2in, cdef2c=al9c,ualantdedαv=al5uepeorf cent level of significance F is less than its critical (or table) value, the null hypothesis is accepted. Hence we conclude that the prices of a commodity in three cities have no significant difference. www.cuidol.in Unit-1(MAP-607) 25 All right are reserved with CU-IDOL

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27 EXTRA Source- Steps for testing Null Hypothesis www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

28 ▸ Step 2: Calculate total variation If a single sample of size n is taken from the population, then estimate of the population variance based on the variance of sampling distribution of mean is given by www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

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31 ▸ Step 3: Calculate variation between sample means This is usually called the ‘sum of squares between’ and measures the variation between samples due to treatments. In statistical terms, variation between samples means is also called the between-column variance. The procedure is as follows: www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

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▸ Step 4: Calculate variation within samples This is usually called 34 the ‘sum of squares within’ and measures the difference within samples due to chance error. Such variation is also called within sample variance. The procedure is as follows: www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

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REFERENCES 45 www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL