Short Answer

1. Explain why the regression line is the 'Best Fit Line' for the data
2. Please put the following correlation coefficient values with the diagrams below.
   \(\mathrm{r}=+0.19 ; \mathrm{r}=+0.95 ; \mathrm{r}=+0.58 ; \mathrm{r}=-0.53 \text { (4 points). }\)
3. Give me one example where causation might be drawn (or has been in your experience) from correlation and whether and why it should have been or not.
4. When and how can adding a variable to an experiment make an \(\mathrm{F}\) value statistically significant?
5. What is regression toward the mean? Give me one example not discussed in lecture or in the book.

Problems by Hand

1. A psychology instructor asked each student to report the number of hours he or she spent preparing for an exam. In addition, the instructor recorded the number of errors made on each student's exam. The data are as follows: (23 points total)

1. Draw the scatterplot ( 2 points)
2. Compute the correlation coefficient (6 points)
3. Calculate the regression equation for predicting \(\mathrm{Y}\) from \(\mathrm{X}\)
4. Draw the regression line with at least 2 LABELLED points on the line (2 points)
5. How many errors would one make after studying for 3 hours? (2 points)
6. Compute the coefficient of determination ( 1 point)
7. Is \(r\) significantly different than zero at the \(\mathrm{p}=.05\) level of significance? ( 2 points)
8. Write an APA-style conclusion based on your findings ( 3 points)

2. Nurcombe, Howell, Rauh, Teti, Ruoff, and Brennan (1984) conducted an intervention program with mothers of low-birth weight infants (LBW). (It is often difficult to recognize signals from low-birth weight infants, and the program offered training in this domain). One group of mothers received instruction on responding to subtle signals of LBW infants, while another group did not receive such instruction. A third group of mothers of normal-birth weight infants served as a control group. In addition, mothers were divided by education level. The dependent variable scores are scores on a maternal adaptation scale, with lower scores indicating better adaptation. Partial data from the study are shown below. What do the results tell us about birth weight and education?

3. Fifteen fourth-grade children who were having difficulty in math were randomly assigned to three groups: (1) tutoring by a teacher, (2) tutoring by a peer (another student), or (3) no tutoring. Here are there scores on a math test 3 months later:

1. Are there any differences in types of tutoring? (10 points)
2. If so, exactly what are they? ( 9 points)
3. How much variance in math test scores is accounted for by tutoring? (2 points)
4. Please write up your APA style conclusion (5 points)

4. A statistics professor conducts an experiment to compare the effectiveness of two methods of teaching his course. Method I is the usual way he teaches the course: lectures, homework assignments, and a final exam. Method II is the same as method I, except the students receiving method II get 1 additional hour per week in which they solve illustrative problems under the guidance of the professor. Since the professor is also interested in how the methods affect students of differing mathematical abilities, volunteers for the experiment are subdivided according to mathematical ability into superior, average, and poor groups. Five students from each group are randomly assigned to method I and five students from each group to method II. At the end of the course, all 30 students take the final exam. Scores are the number of points received out of a total of 50 possible points. The final exam scores are shown below. (25 points):

5. The scores below are days absent and sales from employees at a large corporation ( 25 points).

1. Draw the scatterplot ( 2 points)
2. Compute the correlation coefficient (6 points)
3. Is \(r\) significantly different than zero at the \(\mathrm{p}=.05\) level of significance? ( 2 points)
4. Calculate the regression equation for predicting \(Y\) from \(X\).
5. Draw the regression line with at least 2 LABELLED points on the line ( 2 points)
6. Predict sales given 15 days absent ( 2 points)
7. Calculate the coefficient of determination (1 point)
8. What could we do to increase ' \(r\) ' (2 points)
9. Write up the results in APA style (3 points)

6. Complete the following summary table for an analysis of variance (ANOVA). Please fill in the blank spots in the table (5 points).

N = 40

K = 4

7. Below is the ANOVA summary table for a simple linear regression. Please fill in the blank spots in the table ( 11 points Total, 5 points for filling out table).

1. What proportion of variation is accounted for by the simple linear regression? (3 points)
2. What is the \(\mathrm{F}\) statistic for the regression? Is this statistically significant and how do you know ( 3 points)

8. The following is an ANOVA summary table comparing 2 levels of one variable and 2 levels of another variable, using a separate sample of \(n=15\) participants in each treatment condition. Please complete the summary table (12 points)

Problem Section Using SPSS

1. A researcher has developed a new test of self-esteem. To evaluate the reliability of the test, the researcher obtains a sample of \(\mathrm{n}=8\) subjects. Each individual takes the test on a Monday morning and then returns 2 weeks later to take the test again. The two scores for each individual are reported below. (12 points)
   
   How reliable is the test?
2. To test whether memory changes with age, a researcher conducts an experiment in which there are four groups of six subjects each. The four groups are listed below. All subjects are in good health and matched in other important variables such as years of education, IQ, gender and motivation. Each subject is shown a series of nonsense syllables (a meaningless combination of three letters such as DAF or FUM) at a rate of one syllable every 4 seconds. The series is shown twice after which the subjects are asked to write down as many of the syllables as they can remember. The number of syllables remembered by each subject is shown below. What conclusions can be drawn?
   
3. Assume that you have just accepted a position as chief scientist for a leading agricultural company. Your first assignment is to make a recommendation concerning the best type of grass to grow in the Pacific Northwest and the best fertilizer for it. You fertilize three different types of grass with two different kinds of fertilizer and determine the number of grass blades per square inch of area where the grass is planted. Your recommendation is based on the density of the grass-the denser the grass is, the better (16 points).
   
4. You want to know if a nurse's absences from work in one month (Y) are correlated with her score on a psychological test of burnout (X). Higher scores = more burnout. What conclusions do you draw from the data below? (12 points)
   
5. The following hypothetical data are scores from a test that measures a person's attitudes towards fate. A high score indicates that the person views fate as being out of his or her control. Low scores indicate that he or she feels directly responsible for what happens. Are there any differences among classes in their attitudes towards fate? (12 points)
   
6. A clinical psychologist is interested in the effect that anxiety has on the ability of individuals to learn new material. She is also interested in whether the effect of anxiety depends on the difficulty of the new material. An experiment is conducted in which there are three levels of difficulty (high, medium, and low) and three levels of difficulty (high, medium, and low) for the material that is to be learned. Out of a pool of volunteers, 15 low anxious, 15 medium anxious, and 15 high anxious subjects are selected and randomly assigned 5 each to three difficulty levels. Each subject is given one-half hour to learn the new material, after which the subjects are tested to determine the amount learned. The following data are collected (16 points):

Solution: Factorial ANOVA needs to be used to test for the effect of the difficulty of material and anxiety level of the amount of material learned. The significance of the interaction term will be also tested. The following is obtained with SPSS:

The interaction term Difficulty * Anxiety is significant ( F( 4 , 36 ) = 3 . 58 8, p = . 015 < .05 , \({{\eta }^{2}}=0.285\) ), and both of the main effects of Difficulty ( F( 2 , 36 )= 8 9 . 731 , p < .0 01 , \({{\eta }^{2}}=0.833\) ) and Anxiety ( F( 2 , 36 ) = 6.686 , p = .00 3 < .05 , \({{\eta }^{2}}=0.271\) ).

Hence, the main effects are not interpretable because the effect of each factor is significant and it depends on the level of the other factor.

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