Question 1: A certain type of artificial heart has four independent, critical components. Failure of any of these components is a serious problem. Suppose that 5-year failure rates (expressed as the proportion that fail within 5 years) for these components are known to be:
Component 1 0.01
Component 2 0.06
Component 3 0.04
Component 4 0.02
What is the probability that an artificial heart functions for 5 years?
Question 2: Consider babies born in the 'normal' range of 37-43 weeks gestational age. Extensive data support the assumption that for such babies born in the United States, birth weight is normally distributed with a mean of 3432 grams and standard deviation of 482 grams (Are Babies Normal? The American Statistician [1999] 298-302).
a. What is the probability that the birth weight of a randomly selected baby of this type exceeds 4000 grams? Is between 3000 grams and 4000 grams?
b. What is the probability that the birth weight of a randomly selected baby of this type is either less than 2000 grams or greater than 5000 grams?
c. What is the probability that the birth weight of a randomly selected baby of this type exceeds 7 pounds?
d. How would you characterize the most extreme 0.1% of all birth weights?
e. If x is a variable with a normal distribution and a is a numerical constant (a≠0), then y = ax also has a normal distribution. Use this formula to determine the distribution of birth weight expressed in pounds (shape, mean and standard deviation) and then recalculate part c. How does this compare to your previous answer?
Question 3: Do children diagnosed with attention deficit/hyperactivity disorder (ADHD) have smaller brains than children without this condition? This question was the topic of a research study described in the paper Developmental Trajectories of Brain Volume Abnormalities in Children and Adolescents with Attention-Deficit/Hyperactivity Disorder (Journal of the American Medical Association [2002]: 1740-1747. Brain scans were completed for 152 children with ADHD and 139 children of similar age without ADHD. Summary values for total cerebral volume (in ml) are given in the following Table:
|
|
N
|
Mean
|
Standard Deviation
|
|
Children with ADHD
|
152
|
1059.4
|
117.5
|
|
Children without ADHD
|
139
|
1104.5
|
111.3
|
Do these data provide evidence that the mean brain volume of children with ADHD is statistically significantly smaller than the mean for children without ADHD? Set up a testing protocol, justify the statistical test you have chosen and list all the assumptions you made in using this test/approach. Conduct the test and report your findings.
Question 4: To assess the impact of oral contraceptive use on bone mineral density (BMD), researchers in Canada carried out a study comparing BMD for women who had used oral contraceptives for at least 3 months to BMD for women who had never used oral contraceptives (Oral Contraceptive Use and Bone Mineral Density in Pre- Menopausal Women, Canadian Medical Association Journal [2001]: 1023-1029). Data on BMD in grams per centimeter consistent with summary quantities given in the paper are as follows (the sample size in the actual paper is much larger):
|
Never Used Oral Contraceptives
|
Used Oral Contraceptives
|
|
0.82
|
0.94
|
|
0.94
|
1.09
|
|
1.31
|
0.97
|
|
0.94
|
0.98
|
|
1.21
|
1.14
|
|
1.26
|
0.85
|
|
0.96
|
1.30
|
|
1.09
|
0.89
|
|
1.13
|
0.87
|
|
1.14
|
1.01
|
Assuming the data collection is valid (random sample and that 3 months on oral contraceptives is representative of the population using oral contraceptives), can it be concluded at the 0.05 level of significance that women using oral contraceptives have a lower BMD than women who do not use oral contraceptives? Set up a testing protocol, justify the statistical test you have chosen and list all the assumptions you made in using this test/approach. Conduct the test and report your findings.
Question 5: Five different treatments of fertilizer were applied to a number of plots of corn. Treatment 1 was applied to four plots, Treatment 2 and 3 were applied to six plots, Treatment 4 was applied to seven plots, and Treatment 5 was applied to three plots. The yields per acre are shown in the table
|
Treatment 1
|
Treatment 2
|
Treatment 3
|
Treatment 4
|
Treatment 5
|
|
78.9
|
63.5
|
79.1
|
87.0
|
75.9
|
|
72.3
|
74.1
|
90.3
|
91.2
|
77.2
|
|
81.1
|
75.5
|
85.6
|
75.3
|
81.5
|
|
85.7
|
80.8
|
81.4
|
79.4
|
|
|
|
71.3
|
74.5
|
80.7
|
|
|
|
79.4
|
95.3
|
82.8
|
|
|
|
|
|
89.6
|
|
a. Test the hypothesis that the mean yield of corn is the same for all fertilizers at a 0.05 level of significance. Set up a testing protocol, justify the statistical test you have chosen and list all the assumptions you made in using this test/approach. Conduct the test and report your findings.
b. How many planned comparions could be set up using this data?
c. Suppose you wished to test Treatments 1 and 2 againist Treatments 3-5 as a planned comparison. Could you also test Treatment 2 against Treatment 4? Explain your response.
Question 6: A study was conducted to determine standard reference values for musculoskeletal ultrasonography in healthy adults. Independent random samples of men and women were obtained and the sagittal diameter (in mm) of the biceps tendon was measured in each subject. The resulting summary statistics are given in the following table:
|
Group
|
Sample Size
|
Sample Mean
|
Sample Standard Deviation
|
|
Women
|
54
|
2.5
|
0.49
|
|
Men
|
48
|
2.8
|
0.49
|
a. Is there any evidence to suggest that the population mean sagittal diameter of women's biceps tendons is different from that found in men? Use α = 0.01.
b. Construct a 95% confidence interval for the difference in population mean sagittal diameters, μ1- μ2
c. What assumptions did you make to complete your analyses?
Question 7: A pharmaceutical company claims that its new vaccine is 90% effective, but the FDA suspects that it is only 40% effective. Devise a procedure to test its effectiveness and use it to find the probability that:
a. the FDA will incorrectly grant the company claim (when the effectiveness is in fact 40%); and
b. the company claim will be incorrectly denied (when the effectiveness is in fact 90%).
8. A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighborhood are sick with the flu (F) while 10% are sick with the measles (M). For simplicity's sake, let's assume the two illnesses are mutually exclusive conditions.
a. A well-known symptom of measles is a rash, denoted R. The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flu also develop a rash, with conditional probability of 0.08.
b. Upon examining the child, the doctor finds a rash. What is the probability that the child has the measles?
Question 9: The active ingredient in antiseptic liquid bandages is 8 hydroxyquinoline (8h). The chemical can be harmful though simple skin contact, by swallowing or inhalation. Suppose the list of ingredients on a 3M Nexcare liquid bandage indicates that the volume of 8h is 1%. A random sample of eight bottles of this product was obtained and each was analyzed to obtain the volume of 8h. The sample mean was 1.025%. The standard deviation of the underlying population was 0.04. Is there any evidence to suggest the true percentage of 8h is different from 1%. Use an α = 0.01.
Question 10: An experimenter produced two kinds of lesions in monkeys - Lesion 1 in one group and Lesion 2 in another group. Then she observed each animal for one month after surgery and rated the amount of aggression each animal displayed. She theorized that the two groups would differ primarily in the variability of the behaviors displayed. The data is as follows:
Lesion 1 Lesion 2
Mean = 95.3 Mean = 94.8
s2 = 13.7 s2 = 6.13
N = 31 N= 31
Do the groups differ in variability? Assume that the scores are normally distributed and use a 0.02 level of significance.