Analysis of Variance

Analysis of Variance

[pic 1]

[pic 2]

1        Statistics for Business and Management

Analysis of Variance

[pic 3]

The value of the measured variable

The population mean

The deviation from the population mean due to systematic factor (negative or positive)

The deviation from the population mean due to random factors (negative or positive

Analysis of Variance

means.

[pic 4]

Analysis of variance can be used to test for the equality of k population

[pic 5]

[pic 6]

[pic 7]

[pic 8]

[pic 9]

SSB: sum of squares due to treatment SSW: sum of squares due to error SST: total sum of squares

3        םיקסע להנמב םיטסיטטס םילדומ

Analysis of Variance

[pic 10]

If the sample statistic F is greater than the critical value in the F table [pic 11] We can reject H0.

[pic 12]

[pic 13]

Value- we find the level of significance for the sample statistic F , if it less than [pic 14] , we can reject H0.4        םיקסע להנמב םיטסיטטס םילדומ

Example 1

In a  study that aims to determine the logo for a new brand, 3 logos were tested on different participants. Each participant rated his purchase intentions for the new brand. Purchase intentions scale ranged from 1- I will surely not buy this brand, to 10- I will surely buy this brand. This is the collected data:

1st logo- purchase intentions: 1, 2, 3

2nd logo- purchase intentions: 4, 5, 6

3rd logo- purchase intentions: 7, 8, 9

Use 0.05 significance level to test whether the logo type has an effect on the purchasing intentions.

[pic 15]

[pic 16]

5        Statistics for Business and Management

Example 1

[pic 17]

[pic 18]

The sample statistic (27) is greater the critical value (5.14). We can reject H0.

The level of significance of the sample statistic (27) is 0.0001, which is less than 0.05, we can reject H0.

[pic 19]

[pic 20]

[pic 21]

7        Statistics for Business and Management

Example 2

A study aims to determine the homepage website of company X in order to attract users to explore it longer used an experiment where 3 versions were shown to different participants and time was measured. This is the collected data:

Version 1- minutes spent: 1, 4, 7

Version 2- minutes spent: 5, 8, 11

Version 3- minutes spent: 9, 12, 15

1. Is there a significant difference between the 3 versions? (assume 5% significance level). Is there a significant difference between the 3 versions? (assume 1% significance level).
2. The researcher decided to test his hypotheses using seconds instead of minutes. Will the variance estimates change? Will F change? Will the significance level change?

Example 2

[pic 22]

The level of significance of the sample statistic (5.33) is 0.0467, which is less than[pic 23]

[pic 24]

0.05 but not less than 0.01, we can only reject H0 at a significance level of 5%.Example 2

9        Statistics for Business and Management

[pic 25]

Example 2

1. The researcher decided to test his hypotheses using seconds instead of minutes. Will the variance estimates change? Will F change? Will the significance level change?

When we multiply our variable by a constant, the variance increases by the squared constant, so

the variance estimates will increase by 3,600 times (60 squared).

The two variance estimates (MSB ,MSW)will increase by 3,600 times, so the sample statistic F will not change. If F doesn’t change, the significance doesn’t change.

A change in the units of measurements doesn’t influence the significance level.

[pic 26]

[pic 27]

11        Statistics for Business and Management

Example 2

1. It turns out that there was a mistake in the data. This is the corrected data: Version 1- minutes spent: 3, 4, 5

Version 2- minutes spent: 7, 8, 9

Version 3- minutes spent: 11, 12, 13

Is there a significant difference between the 3 versions? (assume 5% significance level)

Example 2

1. It turns out that there was a mistake in the data. This is the corrected data: Version 1- minutes spent: 3, 4, 5

Version 2- minutes spent: 7, 8, 9

Version 3- minutes spent: 11, 12, 13

Is there a significant difference between the 3 versions? (assume 5% significance level)

The variance within the groups is smaller, while the variance between the groups is the same. We can say for sure that that there is a significant difference between the 3 versions.