Review questions for the final exam. The actual exam might have additional or different questions, but the questions here are a highly representative sample of what will be on the final.

Exam 5 (Final). K300 Statistics, Prof. Kruschke. Tuesday May 3, 2005, 12:30-2:30pm

Be sure to (i) write your name on every page, (ii) write clearly, (iii) show your work --an unannotated sequence of numbers that mysteriously ends up with the correct numerical answer will not  be given full credit-- and (iv) answer every part of every question.

Reminder of formulas:

Assuming equal variances, the t for two independent groups is

where .

For single factor (i.e., one way) ANOVA, or for main effects of multi-factor ANOVA,

where .

Tables for critical t and F values are displayed at the front of the room.

1. Suppose we have scores in a population that is uniformly distributed from a minimum value a  to a maximum value b, with standard deviation s.

Draw a graph of the distribution of the sample mean when N=1. Label both axes, show tic marks on the x-axis, label the shape of the distribution, and label its standard deviation (i.e., standard error, sM) in terms of s.

Draw a graph of the distribution of the sample mean when N=100. Label both axes, show tic marks on the x-axis, label the shape of the distribution, and label its standard deviation (i.e., standard error, sM) in terms of s.

[The actual exam will have space after each question for writing answers.]

2. State the definition of statistical power.

3. Suppose you’re doing an experiment with four groups that vary on a single factor, and you’re going to analyze the data with a one way ANOVA. What are three realistic ways (i.e., the three ways emphasized in homework) to increase statistical power? Be careful to state your answer so it addresses this case with more than two groups.

4. Suppose a researcher runs an experiment with equal numbers of subjects in each of two groups. The researcher reports that “t(46) = 2.13, p<.05 (two tailed).”

How many subjects were in each group?

State in words exactly what “p<.05 (two tailed)” means.

5. You have used tables that list the critical values for t and F. The critical value for F(2,27) is 3.36, at the .05 significance level.

If you are using that critical value to analyze a single-factor ANOVA, how many groups are there?

If there are an equal number of subjects per group, how many subjects per group are there?

What assumptions went into creating that critical value? That is, what are the three crucial properties of the population distributions that were used to generate the critical value?

6. A researcher predicts that two drugs will interact, such that each drug individually enhances immune system function a little, but the two drugs together will enhance immune system function much more than the sum of the two individual influences. The researcher uses a 2 x 2 design (two placebos, one placebo and one drug, one placebo and the other drug, and both drugs) and puts 500 people at random in each of the four groups, for a total of 2,000 subjects. The researcher finds a significant interaction. What must we be wary of when interpreting this result?

7. A chemical company examines an effect of its sewer waste on a sample of 5 fish, and finds no significant effect in the sample. What must we be wary of when interpreting this result?

8. What is the definition of effect size for two groups? State and explain the defining formula.

What is the definition of effect size for multiple groups in single-factor ANOVA? (State and explain the defining formula; do not  use .)

9. Suppose a researcher has run an experiment involving a single factor with four treatment groups. When using the raw values of the data, the overall ANOVA is not significant, but after taking the square root of all the scores, the overall ANOVA is  significant. Which conclusion should the researcher trust more, and why? That is, what further properties of the data determine which ANOVA to trust?

10. Suppose a researcher runs a two group experiment, and finds that scores in the first group tend to be low but are badly skewed to the right, whereas scores in the second group tend to be higher but are badly skewed to the left.

Is it likely that the researcher could find a transformation that would normalize (un-skew) the data? Why or why not?

Suppose the researcher takes a resampling approach. What would be the population distribution for the null hypothesis and why? Briefly describe what the researcher would then do with that population distribution to test the null hypothesis.

11. Suppose there are N=10 subjects per group in two groups. M1=5.0 with SS1 = 9.0 and M2=7.0 with SS2=16.0. Are the group means significantly different? What is the estimated effect size?

12. Suppose there are N=10 subjects per group in three groups. M1=5.0 with SS1 = 9.0, M2=7.0 with SS2=16.0, and M3=4.0 with SS3=12.25. Are the group means significantly different? What is the estimated effect size?

13. [This one has been substantially changed and clarified after being presented in class.] Consider the two factor experiment results shown below:

Col 1

Col 2

Row 1

Row 2

In the table above, fill in the missing numbers, in the order asked for below:

What is the value of the grand mean, GM?

What are the values of the marginal means for Row 1 and Row 2, M1. and M2.? Using those marginal means and the grand mean, determine the values of the Row1 Effect and Row2 Effect. (Hint: One of the Effects will be positive, the other negative.)

What are the values of the marginal means for Col 1 and Col 2, M.1 and M.2? Using those marginal means and the grand mean, determine the values of the Col1 Effect and Col2 Effect. (Hint: One of the Effects will be positive, the other negative.)

Now determine the Interaction effects in the four central cells of the table, by writing the values of the row and column effects in the blanks and determining the remainder.

If an ANOVA were conducted on the raw data that produced those cell means, could the row effect be significant? Why or why not?

Suppose an ANOVA were conducted on the raw data that produced those cell means, and it showed that the interaction is significant. Explain the meaning of the interaction in terms of the row and column effects.