# Statistics: ANOVA, F-ratio, t-test, etc

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1) Describe the similarities between an F-ratio and a t statistic?

2) Explain why you should use ANOVA (analysis of variance) instead of several t tests to evaluate mean differences when an experiment consists of three or more treatment conditions.

3) A research study comparing three treatment conditions produced means of M^1 = 2, M^2 = 4, and M^3 = 6.

a) How many treatment conditions were compared in the study?

b) How many individuals participated in the entire study?

5) The following summary table presents the results from an ANOVA comparing three treatment conditions with n = 8 participants in each treatment. Complete all missing values. (Hint: Start with the df column.)

[see attachment for table]

6) A common science fair project involves testing for the effects of music on the growth of plants.

1) Explain the advantage of using a two factor design instead of using two single factor designs (one for each of the two factors).

2) The following matrix presents the results of a two factor experiment with two levels of factor A, two levels of factor B, and n = 10 subjects in each treatment condition. Each value in the matrix is the mean score for the subjects in that treatment condition. Notice that one of the mean values is missing

A) What value should be assigned to the missing mean so that the resulting data would show no main effect for factor A?

B) What value should be assigned to the missing mean so that the data would show no interaction?

#### Solution Preview

Chapter 13 ^ means to power

1) Describe the similarities between an F-ratio and a t statistic?

You use an F-ratio when doing ANOVA. ANOVA can be used to compare the means of more than 2 groups. Since t-tests can be used to compare the means of only two groups, you can consider the F-ratio/ANOVA a broader form of the t-statistic/t-test.

If you use an F-ratio when you could have also used a t-statistic (comparing two groups, in a linear regression analysis, etc.), the F-ratio will be equal to the t-statistic squared.

2) Explain why you should use ANOVA (ANALYSI OF Variance) instead of several t tests to evaluate mean differences when an experiment consists of three or more treatment conditions.

Because if you do multiple t-tests, you will increase the chances of reporting a false positive (i.e. a Type I error). If you are using a 0.05 level of significance, you have a 5% chance of getting a Type I error in each test.

If you do several t-tests, each of them has a 5% chance of getting a Type I error, so overall, your chance of getting a Type I error is much higher than you wanted. If you do an ANOVA, you are only doing one test, so you can keep your risk of a Type I error where you want it.

3) A research study comparing three treatment conditions produced means of M^1 = 2, M^2 = 4, and M^3 = 6.

a) Compare the variance for the set of three means. (Treat the means as a sample of n=3 values and compute the sample variance.)

The variance of the list 2, 4, 6 is equal to 4.

b) Now we will change the third mean from M^3 = 6 to M^3 = 15. Notice that we have substantially increased the difference among the three means. Compute the variance for the new set of n = 3 means. You should find that the variance is much larger than the value obtained in part a. Note: the variance provides a measure of the size of the mean differences.

The variance of the list 2, 4, 15 is equal to 49.

4) The results from an independent measures research study ...