# Project #49659 - Statistics

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1. When doing an experiment with many groups, what is the problem with doing t tests between all possible groups without any correction? Why does use of the analysis of variance avoid that problem?

2. What are the assumptions underlying the analysis of variance?

3. Find Fcrit for the following situations:

a. df(numerator) =2, df(denominator) =16, α = 0.05

b. df(numerator) =3, df(denominator) =36, α = 0.05

c. df(numerator) =3, df(denominator) =36, α= 0.01

What happens to Fcrit as the degrees of freedom increase and alpha is held constant? What happens

to Fcrit when the degrees of freedom are held constant and alpha is made more stringent?

4. For each of the variables identified in Question 12, state how power is affected if the variable is increased. Use the equation for Fobt on p. 421 to justify your answer.

5. Assume you are a nutritionist who has been asked to determine whether there is a difference in sugar

content among the three leading brands of breakfast cereal (brands A, B, and C). To assess the amount

of sugar in the cereals, you randomly sample six packages of each brand and chemically determine their sugar content. The following grams of sugar were found:

Breakfast Cereal

A        B           C

1        7            6

2        5            4

3        3            4

3        7            5

2        4            7

6        7            8

a. Using the conceptual equations of the one-way ANOVA, determine whether any of the brands differ

in sugar content. Use α = 0.05.

b. Same as part a, except use the computational equations. Which do you prefer? Why?

c. Do a post hoc analysis on each pair of means using the Tukey HSD test with α = 0.05 to determine which cereals are different in sugar content.

d. Same as part c, but use the Scheffe test.

e. Explain any differences between the results of part c and part d.

6. Assume you are employed by a consumer-products rating service and your assignment is to assess car batteries. For this part of your investigation, you want to determine whether there is a difference in useful life among the top-of-the-line car batteries produced by three manufacturers (A, B, and C). To

provide the database for your assessment, you randomly sample four batteries from each manufacturer and run them through laboratory tests that allow you to determine the useful life of each battery. The following are the results given in months of useful ,battery life:

Battery Manufacturer

A     B      C

56      46        44

57      52        53

55      51        50

59     50         51

a. Use the analysis of variance with α =0.05 to determine whether there is a difference among these three brands of batteries.

b. Suppose you are asked to make a recommendation regarding the batteries based on useful life. Use the Tukey HSD test with α =0.05 to help you with your decision.

7. A university researcher knowledgeable in Chinese medicine conducted a study to determine whether

acupuncture can help reduce cocaine addiction. In this experiment, 18 cocaine addicts were randomly

assigned to one of three groups of 6 addicts per group. One group received 10 weeks of acupuncture

treatment in which the acupuncture needles were inserted into points on the outer ear where stimulation

is believed to be effective. Another group, a placebo group, had acupuncture needles inserted

into points on the ear believed not to be effective. The third group received no acupuncture treatment; instead, addicts in this group received relaxation therapy. All groups also received counseling over the

10-week treatment period. The dependent variable was craving for cocaine as measured by the number

of cocaine urges experienced by each addict in the last week of treatment. The following are the results.

Acupuncture +Counseling              Placebo +Counseling                  Relaxation Therapy +Counseling

4                                                                       8                                                                        12

7                                                                      12                                                                         7

6                                                                      11                                                                         9

5                                                                       8                                                                          6

2                                                                      10                                                                       11

3                                                                       7                                                                          6

a. Using α = 0.05, what do you conclude?

b. If there is a significant effect, estimate the size of effect, using Åµ2.

c. This time estimate the size of the effect, using η².

d. Explain the difference in answers between part b and part c.

8. It is theorized that repetition aids recall and that the learning of new material can interfere with the recall of previously learned material. A professor interested in human learning and memory conducts a 2 _ 3 factorial experiment to investigate the effects of these two variables on recall. The material to be recalled consists of a list of 16 nonsense syllable pairs. The pairs are presented one at a time, for 4 seconds, cycling through the entire list, before the first pair is shown again. There are three levels of repetition: level 1, in which each pair is shown 4 times; level 2,in which each pair is shown 8 times; and level 3, in which each pair is shown 12 times. After being presented the list the requisite number of times and prior to testing for recall, each subject is required to learn some intervening material. The intervening material is of two types: type 1, which consists of number pairs, and type 2, which consists of nonsense syllable pairs. After the intervening material has been presented, the subjects are tested for recall of the original list of16 nonsense syllable pairs. Thirty-six college freshmen serve as subjects. They are randomly assigned so that there are six per cell. The following scores are recorded; each is the number of syllable pairs from the original list correctly recalled.

Number of Repetitions

(column variable)

Intervening

Material (row            4 times                         8 times                          12 times

variable)4

Number pairs              10 11                           16 12                                 16 14

12 15                           11 15                                 16 13

14 10                          13 14                                 15 16

Nonsense                     8 7                              11 13                                 14 12

syllable pairs                4 5                                9 10                                  16 15

5 6                                8   9                                  12 13

a. What are the null hypotheses for this experiment?

b. Using α =0.05, what do you conclude? Plot a graph of the cell means to help you interpret the results.

9. Assume 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. To provide the database for your recommendation, having just

graduated summa cum laude in statistics, you decide to conduct an experiment involving a factorial independent groups design. Since there are three types of grass and two fertilizers under active consideration, the experiment you conduct is 2 x 3 factorial, where the A variable is the type of fertilizer and the B variable is the type of grass. In your field station, you duplicate the soil and the climate of the Pacific Northwest. Then you divide the soil into 30 equal areas and randomly set aside 5 for each combination of treatments. Next, you fertilize the areas with the appropriate fertilizer and plant in each area the appropriate grass seed. Thereafter, all areas are treated alike. When the grass has grown sufficiently, you determine the number of grass blades per square inch in each area. Your recommendation is based on this dependent variable. The “denser” the grass is, the better. The following scores are obtained:

Per Square Inch

Fertilizer

Red Fescue            Kentucky Blue                 Green Velvet

Type 1                   14      15                           15     17                                 20          19

16      17                           12     18                                 15          22

10                                   11                                            25

Type 2                11          7                           10        6                                 15         11

11          8                             8      13                                18          10

14                                      12                                            19

a. What are the null hypotheses for this experiment?

b. Using α=0.05, what are your conclusions? Draw a graph of the cell means to help you interpret the results.

10. A sleep researcher conducts an experiment to determine whether a hypnotic drug called Drowson, which is advertised as a remedy for insomnia, actually does promote sleep. In addition, the researcher

is interested in whether a tolerance to the drug develops with chronic use. The design of the experiment

is a 2 _ 2 factorial independent groups design. One of the variables is the concentration of Drowson. There are two levels: (1) zero concentration (placebo) and (2) the manufacturer’s minimum recommended dosage. The other variable concerns the previous use of Drowson. Again there are two levels: (1) subjects with no previous use and (2) chronic users. Sixteen individuals with sleep onset insomnia (difficulty in falling asleep) who have had no previous use of Drowson are randomly assigned to the two concentration conditions, such that there are eight subjects in each condition. Sixteen chronic users of Drowson are also assigned randomly to the two conditions, eight subjects per condition. All subjects take their prescribed “medication” for 3 consecutive nights, and the time to fall asleep is recorded. The scores shown in the following table are the mean times in minutes to fall asleep for each subject, averaged over the

3 days:

Concentration of Drowson

Previous Use                        Placebo                                                     Minimum Recommended Dosage

No previous use                     45       53                                                                     30               47

48       58                                                                     33               35

62       55                                                                     40               31

70       64                                                                     50               39

Chronic users                         47       68                                                                      52              46

52        64                                                                      60              49

55       58                                                                       58              50

62       59                                                                       68              55

a. What are the null hypotheses for this experiment?

b. Using α = 0.05, what do you conclude? Plot a graph of the cell means to help you interpret the

results.

11. A study was conducted to determine whether big-city and small-town dwellers differed in their helpfulness to strangers. In this study, the investigators rang the doorbells of strangers living in New York City or small towns in the vicinity. They explained they had misplaced the address of a friend living in the neighborhood and asked to use the phone. The following data show the number of individuals who admitted or did not admit the strangers (the investigators) into their homes:

into their home                                                                                                   into their home

Big-city dweller                      60                                                                                              90                             150

Small-town dweller                70                                                                                              30                             100

130                                                                                           120                             250

Do big-city dwellers differ in their helpfulness to strangers? Use α = 0.05 in making your decision.

12. A researcher believes that individuals in different occupations will show differences in their ability

to be hypnotized. Six lawyers, six physicians, and six professional dancers are randomly selected for the experiment. A test of hypnotic susceptibility is administered to each. The results are shown in

the next column. The higher the score, the higher the hypnotizability. Assume the data violate the assumptions required for use of the F test, but are at least of ordinal scaling. Using α = 0.05, what is your conclusion?

Condition 1 Lawyers                       Condition 2 Physicians                             Condition 3 Dancers

26                                                      14                                                                  30

17                                                       19                                                                  21

27                                                       28                                                                  35

32                                                       22                                                                  29

20                                                       25                                                                  37

25                                                       15                                                                  34

13. A psychologist investigates the hypothesis that birth order affects assertiveness. Her subjects are 20 young adults between 20 and 25 years of age. There are seven first-born, six second-born, and seven third born subjects. Each subject is given an assertiveness test, with the following results. High scores indicate greater assertiveness. Assume the data are so far from normally distributed that the F test can’t be used, but the data are at least of ordinal scaling. Use α = 0.01 to evaluate the data. What is your conclusion?

Condition 1 First-Born                      Condition 2 Second-Born                               Condition 3 Third-Born

18                                                         18                                                                          7

8                                                         12                                                                        19

4                                                           3                                                                          2

21                                                         24                                                                        30

28                                                          22                                                                        18

32                                                            1                                                                          5

10                                                                                                                                       14

14. A major oil company conducts an experiment to assess whether a fi lm designed to tell the truth

about, and also promote more favorable attitudes toward, large oil companies really does result in more favorable attitudes. Twelve individuals are run in a replicated measures design. In the “Before” condition, each subject fills out a questionnaire designed to assess attitudes toward large oil companies. In the “After” condition, the subjects see the fi lm, after which they fill out the questionnaire. The following scores were obtained. High scores indicate more favorable attitudes toward large oil companies.

Before                         After

43                                  45

48                                  60

25                                  22

24                                  33

15                                    7

18                                  22

35                                  41

28                                  21

41                                  55

28                                  33

34                                  44

12                                  23

Analyze the data using the Wilcoxon signed ranks test with α =0.051 tail. What do you conclude?

15. A social scientist believes that university theology professors are more conservative in political

orientation than their colleagues in psychology. A random sample of 8 professors from the theology department and 12 professors from the psychology department at a local university are given a 50-point questionnaire that measures the degree of political conservatism. The following scores were obtained. Higher scores indicate greater conservatism.

a. What is the alternative hypothesis? In this case, assume a non-directional hypothesis is appropriate

because there are insufficient theoretical and empirical bases to warrant a directional hypothesis.

b. What is the null hypothesis?

c. What is your conclusion? Use the Mann–Whitney U test and α =0.052 tail.

Theology Professors                         Psychology  Professors

36                                                                           13

42                                                                           25

22                                                                           40

48                                                                           29

31                                                                           10

35                                                                           26

47                                                                           43

38                                                                           17

12

32

27

32

 Subject Mathematics Due By (Pacific Time) 12/03/2014 08:00 am
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