Wednesday, 7 March 2018

Hypothesis Testing Problems - Question 10(Sussan Sound predicts that....)

Question on ANOVA
Sussan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides 24 students into three groups of 8 each. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with nose that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. Their scores are tabulated below.

Group1: Constant sound: 7,4,6,8,6,6,2,9
Group 2: Random sound: 5,5,3,4,4,7,2,2
Group 3: No sound at all: 2,4,7,1,2,1,5,5

Learn about Anova and the steps to follow

This is an interesting challenge and we will do it together, just keep a glass of water close by!

The approach we would use
We would use an  aproach that would make is very clear and easy. So you will need MS excel, so we can be faster. You may also do it manually on paper and then verify your results. But to save time, we would use excel.

Step by Step Solution
We need to transfer this data to excel first, which I actually have done and we have the excel table below.

Figure 2: Means of Groups

Learn how to transfer data to excel from a web page without typing.

Step 1: Calculate all the means.
We have done this step using excel as you can see in the table in Figure 1. You could verify this by hand. Watch a video on how to do this

Step 2: Setup the null and alternate hypothesis
You should already know how to do this by now
H0: μ1 = μ2 = μ3
Ha: μ1 ≠ μ2 ≠ μ3

Step 3: Calculate the Sum of Squares
We would extend the excel sheet in Figure 1 to calculate mean squares of the groups. Using the formula we arrive at Figure 2. The formulars are very clear

The relationship between SSB, SSW and SST is given in the formular below:

I have created an excel sheet used to calculate SST, so you can verify. You can also download this excel sheet from here.

Download this sheet

Step 4: Calculate the Degrees of Freedom

dftotal =        n - 1 = 24 - 1 = 23
dfwithin =     n - k = 24 - 3 = 21
dfbetween = k - 1 = 3 - 1 = 2

Verify that dftotal = dfbetween + dfwithin

Step 5: Calculate the Mean Squares
Remember these formulas:

 We calculate the Mean Squares Between(MSB)

Then we calculate the Mean Squares Within(MSW):

 Wow! Breath in, breath out! we are almost there!!

Step 6: Calculate the F statistic
The formula is given below:

At this point, our excel sheet have been extended to calculate all the values. The formula used is written beside some of the values

Our final excel sheet would look like the one below:

 Get This excel sheet from here

Step 7: Look up F from table and state Conclusion
Access statistical tables from here
From the table of F distribution, the critical value of F for 0.05 significance and degrees of freedom of(df1 = 21 and df2 = 2) we have:

F = 3.4668

Since the calculated(absolute value) of F is greater than the tabulated value, we reject the null hypothesis and conclude that at least two of the means are significantly different from each other.

Hurray!!! We are done finally! So go take a bottle of beer! Then take the quiz below
Try another one-way ANOVA problem(Question 11)
Watch the vidoe here for step by step explanation

Simple Quiz
Complete the Anova table below with actual values.
If you do it, let me know in the comment box by the left of this page.