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Statistics: Testing Hypotheses

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The attached file contains the yield for a money market account, a one-year certificate of deposit (CD), and a five-year CD for 23 banks.
a) At the 0.05 level of significance, is there evidence of a difference in the mean yield of the different accounts?
b) If appropriate, determine which accounts differ in mean yields.
c) At the 0.05 level of significance, is there evidence of a difference in the variation in yields among the different accounts?
d) At the 0.05 level of significance, is there evidence of a difference in the median yields of the different account?
e) Compare the results of (a) with (d).

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This solution involves step-by-step solutions for testing hypotheses.

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Statistics : Hypothesis Testing, T-tests, One-Tailed and Two-Tailed Tests (12 Problems)

For each of the following problems, be sure to specify the null hypothesis being tested, and whether you will use a t test for independent samples or a t test for nonindependent samples; also, specify whether you will use a one-tailed or two-tailed test.

1. In a study designed to discover whether men or women drink more coffee, a researcher (working on a very limited budget) observes five men and five women randomly selected from her university department. Here's what she found:

Number of Cups of Coffee
In 1 Day at Work

Men Women
5 8
1 3
4 7
2 3
3 5

(a) Run the appropriate test, assuming that both men and women were originally part of one random sample, with n = 10, and were then divided into men's and women's groups.
(b) Use the same data, but this time assume that the men were randomly selected and then women were selected so that each man could be matched with a woman of the same age, job classification, and overall health status.

2. Using hospital and agency records, you locate six pairs of identical twins, one of whom was adopted at birth and the other of whom was in foster care for at least 3 years. All the twins are now 5 years old. You want to show that early adoption leads to better intellectual ability, so you test all the twins with the Wechsler Intelligence Scale for Children (WISC). Your results are as follows:

WISC Score
Twin Pair No. Adopted Twin Foster-Care Twin
1 105 103
2 99 97
3 112 105
4 101 99
5 124 104
6 100 110

3. The following table contains scores on an index of depression for three groups of clients at a college counseling center. Group 1 clients have received six sessions of counseling; group 2 clients were put on a waiting list for 6 weeks and asked to keep a personal journal during that time; group 3 clients were put on the waiting list with no other instructions. Use a t test to decide whether:
(a) group 2 (journal) clients scored differently from group 3 (control) clients.
(b) Group 1 (counseled) clients scored differently from group 2 (journal) clients.
(c) Group 1 (counseled) clients scored higher than group 3 (control) clients.

Group 1 Group 2 Group 3
(counseled) (journal) (control)
22 6 8
16 10 6
17 13 4
18 13 5
8 2
4. A researcher tests that high-frequency hearing acuity of a group of teens 2 days before they attend a rock concert; 2 days after the concert, she tests them again. Here are her results; she hopes to show that the teens hear better before the concert than afterward (the higher the score on this test, the poorer the hearing).

Subject Scores Scores
(Preconcert) (Postconcert)
Tom 12 18
Dan 2 3
Sue 6 5
Terri 13 10
Karen 10 15
Lance 10 15
Christy 5 6
Jan 2 9
Lenora 7 7
Roberta 9 9
Dave 10 11
Victoria 14 13

5. Cindy, who has had several autumn outings spoiled by bad weather, is convinced that it rains more on weekends than on weekdays in the fall. To test this hypothesis, she randomly selects 10 weekdays and 10 weekend days from the last fall, and finds out how much rain fell on each. Her data follow. (With such small numbers, roundoff errors can make a big difference. Best carry your work in this problem out to four places, instead of the usual two.)

Rainfall on Weekdays Rainfall on Weekend Days
0 .5
.3 0
.2 0
0 0
.15 .2
0 0
0 .1
0 .07
.05 0
.03 .12

6. Do dogs who are fed twice a day eat more in the morning or more in the evening? Here are the data from 15 healthy pets; what do you conclude?

Dog Morning Feeding (oz) Evening Feeding (oz)
Rover 5.9 5.8
Spot 9.9 7.6
Zachary 1.3 1.9
Cagney 8.6 7.2
Clem 7.1 7.3
Claudia 6.0 4.2
Johann S. Bark 7.3 7.2
Tigger 3.3 3.2
Beelzebub 11.9 10.0
Whiskers 5.4 5.2
Him 8.8 9.0
Chiggers 6.9 6.3
Sam 6.5 6.5
Lucy 5.4 5.1
Frisky 5.8 7.2

7. You really are interested in dogs, so you decide to test another hypothesis: Male dogs generally eat more than females. In the data in Problem 6, Rover, Spot, Zachary, Clem, Johann Bark, Beelzebub, Him, and Sam are males. Is your hypothesis supported?

8. The registrar at Cow Catcher College is interested in the relationship between academic achievement and early choice of major. She randomly selects 20 students from the sophomore class at CCC, and records their GPAs and whether they've decided on a major yet. Given the following data, what can she conclude?

GPA When GPA When
Major Is Chosen Major Not Chosen
3.5 3.2
3.1 2.5
2.9 2.0
4.0 3.7
3.8 1.5
3.2 2.9
3.7 3.4
3.5 3.3

1. A researcher is interested in differences among blondes, brunettes, and redheads in terms of introversion/extroversion. She selects random samples from a college campus, gives each subject a test of social introversion, and comes up with the following:

Blondes Brunettes Redheads
5 3 2
10 5 1
6 2 7
2 4 2
5 3 2
3 5 3

Use a simple ANOVA to test for differences among the groups.

2. A (hypothetical!) study of eating patterns among people in different occupations yielded the following:

Bus Drivers College Professors U.S. Presidents
(N = 10) (N = 10) (N = 4)
Mean junk food score 12 17 58.3

Source df Sum of squares Mean square F

Between 2 6505 3252.5 6.1**
Within 21 11197 533.2

(a) What do you conclude?
(b) Perform the appropriate post-hoc analysis.

3. Farmer Hensh suspects that his chickens like music, because they seem to lay more eggs on days when his children practice their band instruments in the hen house. He decided to put it to a scientific test, and records the number of eggs collected each day for a month, along with the music provided on that day:

No Music Mary (Piccolo) Benny (Clarinet) Satchmo (Trumpet)
11 12 35 52
26 17 42 78
31 19 31 16
18 25 33 41
15 32 44 25
27 40 55
30 57
What does he conclude, and how does he explain it?

4. The college librarian at Harton University has been keeping track of who checks out library books. After 6 months of collecting data, he performed an ANOVA, scribbling his results on a sheet of paper. Unfortunately, his dog got into his brief-case and what you see below is all that could be resurrected. Assuming that his calculations were accurate, what could you conclude from his research? Draw a graph of the group means to make your explanation clear:

This is the librarian's sheet of paper:

Compare # of books used by Math majors, psych majors, education majors. Divide each into graduate students (G) and undergraduate (U).

Group means look good __

Math Psych Education
Undergrad. 15 13 7
Grad. 8 42 26

Source df SS MS f
Major 2 ___ ___ 7.82
Level (u/g) 1 ___ ___ 8.33
Interaction (MxL) 2 ___ ___ 9.77
Within Groups 344 ___ ___ ___

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