Are some methods of washing hands more effective at removing bacteria than others? To look at this question, a student tested four different methods - washing with water only, washing with regular soap, washing with antibacterial soap (ABS), and spraying hands with antibacterial spray (AS) (containing 65%ethanol as an active ingredient).

The student suspected that the number of bacteria on her hands before washing might vary considerably from day to day. To help even out the effectsof those changes, she generated random numbers to determine the order ofthe four treatments. Each morning, she washed her hands according to thetreatment randomly chosen. Then she placed her right hand on a sterile mediaplate designed to encourage bacteria growth. She incubated each plate for 2days at 36C, after which she counted the bacteria colonies. She replicated thisprocedure eight times for each of the four treatments. Resulting data are written in Matlab code below (you can copy-paste this to Matlab).

bacteria=[74 84 70 51 135 51 164 5 102 110 88 19 124 67 111 18 105 119 73 58 139 108 119 50 170 207 20 82 87 102 95 17];

method=categorical({'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol' 'Water' 'Soap' 'Antibacterial' 'Alcohol'})

We would like to have Analysis of Variance (ANOVA) to test whether there is a difference in average bacterial count across methods of washing hands, but first we will check assumptions.

Do these assumptions seem reasonable?

Recall the bacteria data collected previously, where a student compared washing her hands with water only, washing with regular soap, washing with antibacterial soap, and spraying hands with an alcohol spray.She wanted to answer the question: are some methods of washing hands more effective at removing bacteria than others? Now we will complete the Analysis of Variance.

(i)Write down the null and alternative hypotheses, using appropriate notation.

What is theP

-value for this test?(Enter your answer correct to 3 decimal places)

How was thisP

-value calculated?Include in your answer the null distribution of the test statistic and a probability expression for theP

-value as a function of the test statistic.

What would you conclude from the output?Relate your answer to the original research question.