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UCLA Statistics, Neurology, LONI

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STAT 13	*Introduction to Statistical Methods for the Life and Health Science*
Department of Statistics	Instructor: Ivo Dinov
Homework 6	Due Date: Wednesday, Nov. 21, 2007

Please, submit your homework before lecture on the due date. See the HW submission rules. On the front page include the following header.

(HW_6) Do these problems on pages 226-255, Samuels & Witmer's text Statistics for the Life Sciences, Prentice Hall (2003).

7.8: Two varieties of lettuce were grown for 16 days in a controlled environment. The table shows the total dry weight (in g) of the leaves of nine plants of the variety Salad Bowl and six plants of the variety Bibb. Compute the SE(y^-₁ - y^-₂) for these data.

y^-	3.259	1.413
Index	Salad Bowl	Bibb
1	3.06	1.31
2	2.78	1.17
3	2.87	1.72
4	3.52	1.20
5	3.81	1.55
6	3.60	1.53
7	3.30
8	2.77
9	3.62
S	0.4	0.220

7.11: Ferulic acid is a compound that may play a role in disease resistance in corn. A botanist measured the concentration of soluble ferulic acid in corn seedlings grown either in the dark or in a light/dark photo period. The results (nmol acid per g of tissue) are shown in the table.

What is a reasonable scientific question to ask for this study and that is the critical population parameter of interest?
Construct a 92% CI(parameter)
Construct a 82% CI (Parameter)

	Sample Mean	Sample Standard Deviation	Sample Size
Dark	92	13	4
Photo period	115	13	4

7.14: Prothrombin time is a measure of the clotting ability of blood. For ten rats treated with an antibiotic and ten control rats, the prothrombin times (in seconds) were reported as follows:

	Antibiotic	Control
N	10	10
y^-	25	23
S	10	8

Construct a 90% confidence interval for the difference in population means. (Assume that the two populations from which the data came are normally distributed.) Note: Formula (7.1) yields 17.2 degrees of freedom for these data.
Interpret the CI in the settings of this problem.

7.16: In a field study of mating behavior in the Mormon cricket (Anabrus simplex), a biologist noted that some females mated successfully while others were rejected by the males before coupling was complete. The question arose whether some aspect of body size might play a role in mating success. The data below summarizes measurements of head width (mm) in the two groups of females.

Construct a 95% confidence interval for the difference in population means.

Interpret your CI in the context of the study.

Successful:	n=22	mean=8.500	S=0.289
Unsuccessful:	n=17	mean=8.441	S=0.262

7.19: Researchers were interested in the short-term effect that caffeine has on heart rate. They took a random sample of individuals and measured each person's resting heart rate. Then they had each subject drink six ounces of coffee. Nine of the subjects were given coffee containing caffeine and eleven were given decaffeinated coffee. After ten minutes each person's heart rate was measured again. The data in the following table show the change in heart rate (beats per minute); a positive number means that heart rate went up and a negative number means that heart rate went down.

Use these data to construct a 90% confidence interval for the difference in mean heart rate change between the caffeinated and decaffeinated coffee groups. You may proceed as though the assumptions have been checked and deemed acceptable. Note: Formula (7.1) yields 17.3 degrees of freedom for these data.
Interpret your CI in this setting.

Subject	Caffeine	Decaf
1	28	26
2	11	1
3	-3	0
4	14	-4
5	-2	-4
6	-4	14
7	18	16
8	2	8
9	2	0
10		18
11		-10
n	9	11
y^-	7.3	5.9
S	11.1	11.2
SE	3.7	3.4

7.21: A researcher investigated the effect of green light, in comparison to red light, on the growth rate of bean plants. The following table shows data on the heights of plants (in inches), from the soil to the first branching stem, two weeks after germination. Assume that the heights follow a normal distribution. Use the conservative degree of freedom df = 16 for the following questions.

Construct a 90% confidence interval for the difference in mean effect red light has on bean plant growth.
Interpret this confidence interval. That it, explain what the numbers in the interval mean.

	Red	Green
N	17	25
y^-	8.36	8.84
s	1.50	1.78

7.27: In a study of the nutritional requirements of cattle, researchers measured the weight gains of cows during a 78-day period. For two breeds of cows, Hereford (HH) and Brown Swiss/Hereford (SH), the results are summarized in the following table. Use a T test to compare the means, with a = 0.10.

	HH	SH
N	33	51
y^-	18.3	13.9
S	17.8	19.1

7.31: In a study of the development of the thymus gland, researchers weighted the glands of 10 chick embryos. Five of the embryos had been incubated for 14 days and five had been incubated for 15 days. The weights of the thymus glands are reported below.

Use the T test to compare the means at a = 0.10.
Not that the chicks that were incubated longer had smaller mean thymus gland. Explain this backwards result!

Index	Thymus Grand Weight 14 days	Thymus Grand Weight 15 days
1	29.6	32.7
2	21.5	40.3
3	28	23.7
4	34.6	25.2
5	44.9	24.2
n	5	5
y^-	31.72	29.22
S	8.73	7.19

7.37: The table contains the number of bacteria colonies present in each of several petri dishes after E. Coli bacteria were added to the dishes and they were incubated for 24 hours. The "soap" dishes contained a solution prepared from ordinary "soap" the "control" dishes contained a solution of sterile water.

Use the T-test to investigate soap affects on the number of bacteria colonies that form (use a = 0.10).
State your conclusions in the context of the study.

Index	Control	Soap
1	30	76
2	36	27
3	66	16
4	21	30
5	63	26
6	38	46
7	35	6
8	45
n	8	7
y^-	41.8	32.4
S	15.6	22.8
SE	5.5	8.6

7.45: Suppose that a 95% CI(µ₁ - µ₂) is calculated to be (-7.4, -2.3). If we test H_o: µ₁ = µ_2
vs.H_a: µ₁ != µ₂ using a=0.10, will we reject H_o? Explain!

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