Business statistics

Lecture Outline

Ø Understand the concept of ANOVA.

Ø Compute and interpret the result of one- way ANOVA.

Analysis of Variance (ANOVA)

Analysis of Variance

Ø Analysis of variance or ANOVA is a technique of testing hypotheses about the significant difference in several population means.

Ø In analysis of variance, the total variation in the sample data can be on account of two components, namely, variance between the samples and variance within the samples.

Ø Variance between the samples is attributed to the difference among the sample means.

Ø Variance within the samples is the difference due to chance or experimental errors.

Figure : Partitioning the total sum of squares of the variation for completely randomized design (one-way ANOVA)

SST (total sum of squares) = SSC (sum of squares between columns) + SSE (sum of squares within samples)

Completely Randomized Design (One-way Anova)

Completely randomized design contains only one independent variable, with two or more treatment levels or classifications.

Applying the F -Test Statistic

Ø In case of ANOVA, F value is obtained by dividing the treatment variance (MSC) by the error variance (MSE).

Ø F test statistic in one-way ANOVA

Ø The F test statistic follows F distribution with k – 1 degrees of freedom corresponding to MSC in the numerator and n – k degrees of freedom corresponding to MSE in the denominator.

The ANOVA Summary Table

Figure : Rejection and non-rejection region (acceptance region) when using ANOVA to test null hypothesis

Vishal Foods Ltd is a leading manufacturer of biscuits. The company has launched a new brand in the four metros; Delhi, Mumbai, Kolkata, and Chennai. After one month, the company realizes that there is a difference in the retail price per pack of biscuits across cities. Before the launch, the company had promised its employees and newly-appointed retailers that the biscuits would be sold at a uniform price in the country. The difference in price can tarnish the image of the company. In order to make a quick inference, the company collected data about the price from three randomly selected stores across the four cities. Based on the sample information, the price per pack of the biscuits (in rupees) is given.

Example

Use one-way ANOVA to analyse the significant difference in the prices. Take 95% as the confidence level.

Example : Continued

Table: ANOVA table

Volkswagon wants to examine the safety of compact cars, midsize cars, and full-size cars. It collects a sample of three for each of the treatments (cars types). Using the data provided below, test whether the mean pressure applied to the driver’s head during a crash test is equal for each types of car. Use α = 5%.

Example

Compact Cars

Midsize Cars

Full-Size Cars

15 25 10 25 25 5 20 35 15

Students were given different drug treatments before revising for their exams. Some were given a memory drug, some a placebo drug and some no treatment. Test whether the performance is difference across the three groups or not. The exam scores (%) are shown below for the three different groups:

Example

Memory Drug Placebo No Treatment

70 37 3 77 43 10 83 50 17 90 57 23 97 63 30

Mean 83.40 50 16.60 Grand Mean 50

THANK YOU!