Statistics

FOR EACH OF THE FOLLOWING QUESTIONS, PLEASE PROVIDE THE INFORMATION THAT IS REQUESTED. THE POINT VALUE OF EACH QUESTION (OR EACH PART) IS INDICATED IN PARENTHESES AT THE END OF THE QUESTION. TOTAL POINT VALUE IS 50. EQUATIONS AND FORMULAS THAT MIGHT BE NEEDED ARE ON THE LAST PAGE OF THIS EXAM. TABLES FOR z, t, and F WILL BE NEEDED – USE AS APPROPRIATE. PLEASE BE SURE TO PROVIDE YOUR CALCULATIONS – SHOW YOUR WORK.

1. In an independent-groups study, a Mediterranean diet is given to 50 people and a Paleo diet is given to 40 different people. Of interest is whether the Mediterranean diet lowers cholesterol levels more so than the Paleo diet. Participants in this study did not differ significantly at the start of the study with respect to their LDL, HDL, or total cholesterol levels, and were randomly assigned to the two treatment conditions. The Mediterranean group had a mean decrease of 12 mg/dl of LDL, with a standard deviation of 3 mg/dl. The Paleo diet group had a mean decrease in LDL of 8 mg/dl, with a standard deviation of 2 mg/dl.

a. For this between-groups study, state an appropriate set of hypotheses using the proper notation, where you are comparing two samples that are believed to be representative of the population. (2)

b. The result of this study yielded t = 1.496. Using the hypotheses you provided above, what conclusion can you make about the difference between these means at the p =.05 level of significance? Please indicate the critical value from the table, as well as the degrees of freedom associated with this study and statistical test. (3)

2. A researcher for a yogurt manufacturer would like to determine whether adult consumers showed any preference between two new yogurt products that the company recently developed, as one will be marketed as “organic” and the other without the organic designation. To start, a small sample of adults was obtained (n = 9), and on Day 1, given the organic yogurt to eat. The next day, the same 9 adults were given the non-organic yogurt to eat, with the same instructions and the same testing conditions. Participants did not know which type of yogurt (organic or nonorganic) they were eating, as both were in identical containers that were white with green lettering. It was found that these yogurt-eaters ate, on average, 6.0 ounces more of the organic yogurt than of the non-organic product when allowed to consume as much as they wanted.

a. Indicate an appropriate non-directional set of hypotheses for this type of repeated measures study. (1)

b. The result of this t-test was t = 2.63. Using the hypotheses you provided above, what conclusion can you make about this study of yogurt eating at the p = .05 level of significance? Please indicate the critical value from the table, as well as the degrees of freedom associated with this study and statistical test. (3)

F-DISTRIBUTION AND ANALYSIS OF VARIANCE

3. When is a one-way analysis of variance (ANOVA) used to analyze data? (1)

4. Assume you are a nutritionist who has been asked to determine whether there is a difference in the sugar content among three brands of popular breakfast cereal (Life, Lucky Charms, and Frosted Flakes). To assess the amount of sugar in these cereals, you randomly sample six packages of each brand and chemically determine the amount of sugar (in grams) per equal serving size in these samples of cereals. The following grams of sugar were found.

LIFE

LUCKY CHARMS

FROSTED FLAKES

5

7

8

1

7

5

4

5

4

3

3

4

3

6

5

2

4

7

a. State the appropriate hypotheses for the analysis of variance. (2)

H0: There is no difference among of sugar content among the three brand of popular breakfast cereal

H1: At least one of the brand have different sugar amount

b. Indicate the number of degrees of freedom for each of the following:

(1) Between groups: (2)

(2) Within groups: (2)

(3) Total degrees of freedom: (2)

c. The result of this analysis yielded an F ratio of 4.77. What do you conclude about the amount of sugar in these three brands of breakfast cereal? Be sure to indicate the critical value from the table, as well as the alpha level you have set. (3)

5. A lot of people consume caffeine in the morning – but does the amount of caffeine have any obvious behavioral effects? One researcher thought that the amount of caffeine consumed (in milligrams) might influence people’s walking, as measured by their stride length – longer strides would allow people to travel further in some given interval of time. An experimental study was done with four groups of people, with participants all of the same height and leg length. One group consumed 12 ounces of a liquid without caffeine, while participants in the other three groups drank 12 ounces of liquid with either 50 mg, 80, mg, and 100 mg of caffeine. All participants drank these beverages at 8 am, then were asked to walk across campus to another location for the next part of the study. The number of strides (steps) that each participant took in a 15-second observation period was recorded. There were 5 participants in each of the 4 groups. Here are the data:

PLACEBO

50 mg CAFFEINE

80 mg CAFFEINE

100 mg CAFFEINE

6

8

12

14

2

6

9

12

2

4

8

15

3

4

6

10

3

4

8

11

a. State an appropriate set of hypotheses for this one-way analysis of variance. (2)

b. Complete the following summary table for this study. (3)

SOURCE OF VARIATION

SUM OF SQUARES

DEGREES OF FREEDOM

MEAN SQUARES

F

Between

50

Within

32

Total

82

c. What is your conclusion? As in the other questions like this, state your alpha level, degrees of freedom used to find the critical value, and the critical value itself from the table. (3)