What methods did we employ in this experiment?

On each trial you rated the happiness evoked by an image of abstract art. There were no right or wrong answers; we were just interested in your interpretation of the art. You made your judgments under three different conditions. One condition was a normal condition, but the other two conditions might have seemed odd. In one condition you were asked to place (or pretend to place) a pen or pencil between your teeth. This is a special condition because the act of holding a pen in your teeth utilizes facial muscles in a way that is similar to when you smile. The experiment is investigating whether making a smiling-type face influences your judgments of happiness in abstract art. A second condition is a kind of control condition where you held (or pretended to hold) a pen or pencil in your lips. This condition is just as odd as the teeth condition, but does not utilize muscles involved in smiling.

The dependent variable is the happiness rating, with 0 indicating the piece of art does not evoke happiness at all and 100 indicating very much happiness. We will use a one-way analysis of variance (ANOVA) to explore whether the happiness ratings are equal or different across the three conditions.

What do we predict participants will do? Why?

The null hypothesis is that the population mean happiness rating is the same for all three conditions:

H0: No pen = Teeth = Lips,

while the alternative hypothesis is that at least one of the means differs from the others

Ha: i j for some i,j.

If the effect exists, you will reject the null hypothesis, and the mean for the pen in teeth condition will be larger than the other means.

Formulas

There are several ways to calculate the terms for an ANOVA. We will use formulas that make explicit the important parts of the approach. These formulas only apply to the case (as here) where there are equal sample sizes, n, for all groups. To calculate the ANOVA, you need to first calculate the mean and variance of scores for each condition. This is done in the first six questions.

Now you compute two estimates of the population variance (ANOVA assumes that each condition has the same variance). The first estimate is called Mean Square Between (MSB) because it derived from the variance between groups. To compute it we first calculate the variance of the means. For three groups the raw score formula is:

2B=3k=1X2k(3k=1Xk)2/32.

This variance is then used to estimate the variance of the population with the following formula:

MSB=ns2B.

where n is the number of scores in each group.

The next step is to compute a second estimate of the population variance, which is called Mean Square Error (MSE). (Sometimes this estimate is called Mean Square Within, because it is based on the variances within each condition.) This is done by pooling the variances within each group:

MSE=3k=1s2k3

The two estimates of population variance are then compared by computing the F ratio:

F=MSBMSE,

which has two degrees of freedom. The first is for MSB, dfnumerator=K-1, where K is the number of groups. The second is for MSE, dfdenominator = N-K where N is the total number of scores across all groups.

Finally, you can calculate the p-value for the F ratio. This is most easily done with an online calculator.

Mean happiness for art with no pen:

(The difference between your answer and the correct value must be less than 0.01.)

Mean happiness for art with the pen in your teeth:

(The difference between your answer and the correct value must be less than 0.01.)

Mean happiness for art with the pen in your lips:

(The difference between your answer and the correct value must be less than 0.01.)

Variance of the means:

(The difference between your answer and the correct value must be less than 0.001.)

Mean Squared Between (MSB) variance:

(The difference between your answer and the correct value must be less than 0.001.)

Variance of happiness for art with no pen:

(The difference between your answer and the correct value must be less than 0.01.)

Variance of happiness for art with the pen in your teeth:

(The difference between your answer and the correct value must be less than 0.01.)

Variance of happiness for art with the pen in your lips:

(The difference between your answer and the correct value must be less than 0.01.)

Mean Squared Error (MSE) variance:

(The difference between your answer and the correct value must be less than 0.01.)

F value:

(The difference between your answer and the correct value must be less than 0.01.)

Check answer

dfnumerator:

(Your answer must exactly equal the correct value.)

Check answer

dfdenominator:

(Your answer must exactly equal the correct value.)

Check answer

p-value:

(The difference between your answer and the correct value must be less than 0.001.)

Check answer

Do you reject the null hypothesis?:

Check answer

Trial Condition Happiness rating

1 No pen 39.0

2 No pen 60.0

3 No pen 5.0

4 No pen 55.0

5 No pen 18.0

6 No pen 75.0

7 No pen 93.0

8 No pen 61.0

9 No pen 61.0

10 No pen 85.0

11 No pen 86.0

12 Pen in teeth 26.0

13 Pen in teeth 35.0

14 Pen in teeth 55.0

15 Pen in teeth 21.0

16 Pen in teeth 35.0

17 Pen in teeth 60.0

18 Pen in teeth 43.0

19 Pen in teeth 64.0

20 Pen in teeth 86.0

21 Pen in teeth 75.0

22 Pen in teeth 18.0

23 Pen in lips 54.0

24 Pen in lips 59.0

25 Pen in lips 75.0

26 Pen in lips 40.0

27 Pen in lips 9.0

28 Pen in lips 86.0

29 Pen in lips 98.0

30 Pen in lips 65.0

31 Pen in lips 56.0

32 Pen in lips 4.0

33 Pen in lips 60.0