Reference no: EM131563

Question 1. Examined the effect of body position on blood pressure, 32 participants had their blood pressures measured while lying down with their arms at their sides and again standing with their arms supported at heart level. 

Blood pressure (mm Hg)
Participant	Recumbent, arm at side		Standing, arm at heart level
BRA	99	71	105	79
JAB	126	74	124	76
FLB	108	72	102	68
VPB	122	68	114	72
MFB	104	64	96	62
EHB	108	60	96	56
GC	116	70	106	70
MMC	106	74	106	76
TJF	118	82	120	90
RRF	92	58	88	60
CRF	110	78	102	80
EWG	138	80	124	76
TFH	120	70	118	84
EJH	142	88	136	90
HBH	118	58	82	58
RTK	134	76	126	68
WEL	118	72	108	68
RLL	126	78	114	76
HSM	108	78	94	70
VJM	136	86	144	88
RHP	110	78	100	64
RCR	120	74	106	70
JAR	108	74	94	74
AKR	132	92	128	88
THS	102	68	96	64
OES	118	70	102	68
RES	116	76	88	60
ECT	118	80	100	84
JHT	110	74	96	70
FPV	122	72	118	78
PFW	106	62	94	56
WJW	146	90	138	94

a.Compute the arithmetic mean and median for the difference in systolic and diastolic blood pressure, respectively, taken in different positions (recumbent minus standing). 

b.Construct two box plots for the difference scores for each type of blood pressure.

c.Based on your answers to parts a and b, comment on the effect of body position on the levels of systolic and diastolic blood pressure.

d.Orthostatic hypertension is sometimes defined based on an unusual change in blood pressure after changing position.  Suppose we define a normal range for change in systolic blood pressure (SBP) based on change in SBP from the recumbent to the standing position in the Table above that is between the upper and lower decile. [decile-each of ten equal groups into which a population can be divided according to the distribution of values of a particular variable.]  What should the normal range be? 

Question 2.Breast cancer is considered largely a hormonal disease.  An important hormone in breast-cancer research is estradiol.  The data below on serum estradiol levels were obtained from 213 breast-cancer cases and 432 age-matched controls.  All women were age 50-59 years. 

Serum estradiol (pg/mL)	Cases (N=213)	Controls (N=432)
1-4	28	72
5-9	96	233
10-14	53	86
15-19	17	26
20-24	10	6
25-29	3	5
30+	6	4

Suppose a serum-estradiol level of 20+ pg/mL is proposed as a screening criterion for identifying breast-cancer cases.

a.What is the sensitivity of this test?

b.What is the specificity of this test?

c.The preceding sample was selected to oversample cases.  In the general population, the prevalence of breast cancer is about 2% among women 50-59 years of age. 

What is the probability of breast cancer among 50-to-59-year-old women in the general population who have a serum-estradiol of ≥  20 pg/mL?  What is another name for his quantity? 

Question3.Suppose 6 of 15 students in a grade-school class develop influenza, whereas 20% of grade-school students nationwide develop influenza. 

a.Is there evidence of an excessive number of cases in the class?  That is, what is the probability of obtaining at least 6 cases in this class if the nationwide rate holds true?

b.What is the expected number of students in the class who will develop influenza?

c.What is the probability of obtaining exactly 6 events for a Poisson distribution with parameter µ=4.0?

d.What is the probability of obtaining at least 6 events for a Poisson distribution with parameter µ=4.0?

e.What is the expected value and variance for a Poisson distribution with parameter µ=4.0?

Question 4.Well-known racial differences in blood pressure exist between white and black adults.  These differences genrally do not exist between white and black children.  Because aldosterone levels have been related to blood-pressure levels in adults in previous research, an investigation was performed to look at aldosterone levels among black children and white children.

a.If the mean plasma-aldosterone level in black children as 230 pmol/L with a standard deviation of 203 pmol/L, then what percentage of black children have levels ≤300 pmol/L if normality is assumed?

b.If the mean plasma-aldosterone level in white children is 400 pmol/L with standard deviation of 218 pmol/L, then what percentage of white children have levels ≤ 300 pmol/L if normality is assumed?

c.The distribution of plasma-aldosterone concentration in 53 white and 46 black children is shown in Figure 5.23.  Does the assumption of normality seem reasonable? Why or why not?  (Hint: Qualitatively compare the observed number of children who have levels ≤ 300 pmol/L with the expected number in each group under the assumption of normality.)

Question 5.There is a sampling distribution of the mean from 200 samples of size 5 from the population of 1000 birthweights.  The mean of the 1000 birthweights is 112.0 oz with standard deviation 20.6 oz. 

a.If the central-limit theorem holds, what proportion of the sample means should fall within 0.5 lb of the population mean (112.0) oz?

b.Answer part a for 1lb rather than 0.5lb. 

c.Compare the results of parts a and b with the actual proportion of sample means that fall in these ranges. 

d.Do you feel the central-limit theorem is applicable for samples of size 5 from this population? Explain.  

Question 6.A study in Pittsburgh measured various cardiovascular risk factors in children at birth and during their first 5 years of life.  In particular, heart rate was assessed at birth, 5 months, 15 months, 24 months, and annually thereafter until 5 years of age.  Heart rate was related to age, sex, race, ad socioeconomic status.  The data were represented relating heart rate to race among newborns. 

Race	Mean heart rate (beats per minute)	sd	n
White	125	11	218
Black	133	12	156

a.Test for the significant difference in mean heart rate between Caucasian and African-American newborns.

b.Report a p-value for the test performed in part a. 

Question 7.A possible important environmental determinant of lung function in children is amount of cigarette smoking in the home.  Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5-9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4L. 

a.What are the appropriate null and alternative hypotheses in this situation?

b.What is the appropriate test procedure for the hypotheses in part a?

c.Carry out the test in part b.

d.Provide a 95% CI for the true mean different in FEV between 5-to9 year old children whose parents smoke and comparable children whose parents do not smoke. 

e.If this is regarded as a pilot study, how many children are needed in each group (assuming equal numbers in each group) to have 95% change of detecting a significant difference using a two-sided test with α=0.05?

f.Answer the question in part e if the investigators use a one-sided rather than a two-sided test.

g.Suppose 40 children, both of whose parents smoke, and 50 children, neither of whose parents smoke, are recruited for the study.

h.How much power would such a study have using a two-sided test with significance level=0.05, assuming that the estimates of the population parameters in the pilot study are correct?

i.Answer part h if a one-sided rather than a two sided test is used. 

Question 8.A common symptom of otitis media in young children is the prolonged presence of fluid in the middle ear, known as middle-ear effusion.  The presence of fluid may result in temporary hearing loss  and interfere with normal learning skills in the first 2 years of life.  One hypothesis is that babies who are breastfed for a least 1 month build up some immunity against the effects of the infection and have less prolonged effusion than do bottle-fed babies.  A small study of 24 pairs of babies is set up, in which the babies are matched on a one-to-one basis according to age, sex, socioeconomic status, and type of medications taken.  One member of the matched pair is a breastfed baby, and the other member is a bottle-fed baby.  The outcome variable is the duration of middle-ear effusion after the first episode of otitis media.

Pair number	Duration of effusion in breastfed baby (days)	Duration of effusion in bottle-fed baby (days)
1	20	18
2	11	35
3	3	7
4	24	182
5	7	6
6	28	33
7	58	223
8	7	7
9	39	57
10	17	76
11	17	186
12	12	29
13	52	39
14	14	15
15	12	21
16	30	28
17	7	8
18	15	27
19	65	77
20	10	12
21	7	8
22	19	16
23	34	28
24	25	20

a.What hypotheses are being test here?

b.Why might a nonparametric test be useful in testing the hypotheses?

c.Which nonparametric test should be used here?

d.Test the hypothesis that the duration of effusion is different among breastfed babies than among bottle-fed babies using a nonparametric test. 

Question 9.Polyunsaturated fatty acids in the diet favorably affect several risk factors for cardiovascular disease.  The principal dietary polyunsaturated fat is linoleic acid.  To test the effects of dietary supplementation with linoleic acid on blood pressure, 17 adults consumed 23 g/day of safflower oil, high in linoleic acid, for 4 weeks.  Systolic blood pressure (SBP) measurements were taken at baseline (before ingestion of oil) and 1 month later, with the mean values over several readings at each visit given in the table below.

Subject	Baseline SBP	1-month SBP	Baseline-1-month SBP
1	119.67	117.33	2.34
2	100.00	98.78	1.22
3	123.56	123.83	-0.27
4	109.89	107.67	2.22
5	96.22	95.67	0.55
6	133.33	128.89	4.44
7	115.78	113.22	2.56
8	126.39	121.56	4.83
9	122.78	126.33	-3.55
10	117.44	110.39	7.05
11	111.33	107.00	4.33
12	117.33	108.44	8.89
13	120.67	117.00	3.67
14	131.67	126.89	4.78
15	92.39	93.06	-0.67
16	134.44	126.67	7.77
17	108.67	108.67	0.00

a.What parametric test could be used to test for the effect of linoleic acid on SBP?

b.Perform the test in part a, and report a p-value.

c.What nonparametric test could be used to test for the effect of linoleic acid on SBP?

d.Perm the test in part c, and report a p-value.

e.Compare your results in part b and d, and discuss which method you feel is more appropriate here. 

Question 10. Gastroenterology

Question 11.Two drugs (A,B) are compared for the medical treatment of duodenal ulcer. For this purpose, patients are carefully matched wit regard to age, sex, and clinical condition.  The treatment results based on 200 matched pairs show that for 89 matched pairs both treatments are effective; for 90 matched papers both treatment are ineffective; for 5 matched pairs drug A is effective, whereas drug B is ineffective; and for 16 matched pairs drug B is effective, whereas drug A is ineffective. 

a.What test procedure can be used to assess the results?

b.Perform the test in part a and report a p-value.  In the same study, if the focus is on the 100 matched pairs consisting of male patients, then the following results are obtained: for 52 matched pairs both treatments are ineffective; for 5 matched pairs drug A is effective, whereas drug B is ineffective; and for 16 matched pairs drug B is effective, whereas drug A is ineffective. 

c.How many concordant pairs are there among the male matched pairs?

d.How many discordant pairs are there among the male matched pairs?

e.Perform a significance test to assess any differences in effectiveness between the drugs among males.  Report p-value. 

Question 12.An investigator wants to study the effect of cigarette smoking on the development of myocardial infarction (MI) in women.  In particular, some question arises in the literature as to the relationship of timing of cigarette smoking to development of disease.  One school of though says that current smokers are at much higher risk than ex-smokers.  Another school of though that says a considerable latent period of nonsmoking is needed before the risk of ex-smokers becomes less than that of current smokers.  A third school of thought is that ex-smokers may actually have a higher incidence of MI then current smokers because they may include more women with some prior cardiac symptoms (e.g., angina) than current smokers.  To test his hypothesis, 2000 disease-free currently smoking women and 1000 disease-free ex-smoking women, age 50-59, are identified in 1996, and the incidence of MI between 1996 and 1998 is noted at follow-up visits 2 years later.  Investigators find that 40 currently smoking women and 10 ex-smoking women have developed the disease.

a.Is a one-sample or a two-sample test needed here?

b.Is a one-sided or a two-sided test needed here?

c.Which of the following test procedure should be used to test the hypothesis? 

• X² test for 2x2 contingency table
• Fisher's exact test
• McNemar's test
• One-sample binomial test
• One-sample t-test
• Two-Sample t test with equal variances

d.Carry out the test procedure (s) mentioned in part c, and report a p-value. 

Question 13.Draw a sample ROC curve.

a.Why would you use an ROC Curve?

b.How would you read an ROC Curve?

Question 14.Assume half of the individuals dying during an interval live the midpoint, one-fourth of the individuals dying during the interval like to the 25^th percentile and one-fourth of the individuals dying during the interval live to the 75^th percentile.