Applied Biostatistics for the Health Sciences. Richard J. Rossi
Under what conditions is the probability of the event “A and B” equal to the product of their respective probabilities?
33 2.33 Suppose that P(A)=0.54,P(B)=0.48, and P(A and B)=0.33. Determinethe probability that the event A does not occur.the probability that the event A or B occurs.the probability that neither event A nor event B occurs.the conditional probability that event A occurs given that the event B will occur.the conditional probability that event B occurs given that the event A will occur.whether or not the events A and B are independent events.
34 2.34 Suppose that P(A)=0.60,P(B)=0.25, and the events A and B are disjoint events. Determinethe probability that event A does not occur.the probability that event A or B occurs.the probability that neither event A nor event B occurs.the conditional probability that event A occurs given that the event B will occur.whether or not the events A and B are independent events.
35 2.35 Suppose that P(A)=0.6,P(B)=0.8 and A and B are independent events. Determinethe probability that the event B does not occur.the probability that the event A and B occurs.the probability that the event A or B occurs.the conditional probability that the event A occurs given the event B will occur.the conditional probability that the event B occurs given the event A will occur.
36 2.36 Of the people who have had a heart attack, suppose that 80% change their diet, 42% get more exercise, and 36% change their diet and get more exercise. Determine the probability that a randomly selected individual who has had a heart attackdoes not get more exercise.changes their diet or gets more exercise.gets more exercise given they change their diet.
37 2.37 In the article “Prevalence and predictability of low-yield inpatient laboratory diagnostic tests” published in JAMA Network Open (Xu, 2019), the authors reported the prevalence, sensitivity, and specificity for diagnosing normal troponin I levels; troponin I is a marker for acute myocardial infarction. The authors reported a prevalence of 0.33, a sensitivity of 0.88, and a specificity of 0.79 for the lab test for troponin I levels. Determine thepositive predictive value (PPV) for this test.negative predictive value (NPV) for this test.
38 2.38 According to the Medscape Today article “Standard care for pap screening” (Lie, 2003), the sensitivity and specificity of the pap smear test are at least 0.29 and 0.97, respectively. If the prevalence of cervical cancer is 0.01, determinethe probability that a woman has a positive test result.the positive predictive value of the pap smear diagnostic test.the negative predictive value of the pap smear diagnostic test.
39 2.39 In the article “Diagnostic testing for Lyme disease: beware of false positives” published in BC Medical Journal (Kling, 2015), the authors reported the sensitivity and specificity for two diagnostic tests, a two-step diagnostic test and a standard laboratory test, shown in Table 2.16. Assuming the prevalence of Lyme disease is 0.01, determineTable 2.16 The Sensitivity and Specificity for Two Tests for Diagnosing Lyme Disease in Exercise 2.40TestSensitivitySpecificityTwo-step0.870.99Lab Test0.700.73the positive predictive value of the two-step diagnostic test.the negative predictive value of the two-step diagnostic test.the positive predictive value of the laboratory diagnostic test.the negative predictive value of the laboratory diagnostic test.
40 2.40 According to the American Red Cross, the percentage of people in the United States having blood type O is 38%. If four people from the United States are selected at random and independently, determine the probability thatnone have blood type O.all four have blood type O.at least one has blood type O.
41 2.41 The autosomal recessive genetic disorder sickle cell anemia is caused by a defect in the hemoglobin beta (HBB) gene. Two defective genes, denoted by SS, are needed for sickle cell anemia to occur in an individual. If each parent carries one sickle HBB gene (S) and one normal HBB gene (A), and a child receives exactly one gene independently from each parent, determinethe probability that a child will have sickle cell anemia.the probability that a child will not have sickle cell anemia.the probability that a child will not have any sickle HBB genes.
42 2.42 What are the four conditions necessary to have a binomial distribution?
43 2.43 Suppose the random variable X has a binomial distribution with n = 10 trials and probability of success p = 0.25. Using the probabilities given in Table 2.17, determineTable 2.17 The Binomial Probabilities for n = 10 Trials and p = 0.25Binomial with n = 10 and p = 0.25xP(X = x)00.05631410.18771220.28156830.25028240.14599850.05839960.01622270.00309080.00038690.000029100.000001the most likely value of X.the least likely value of X.the probability that X is less than 6.the probability that X is greater than equal to 4.the probability that 2≤X≤6.the mean value of X.
44 2.44 Determine the mean, variance, and standard deviation for each of the following binomial distributions.n = 50 and p = 0.4.n = 200 and p = 0.75.n = 80 and p = 0.25.
45 2.45 For what values of p will a binomial distributionhave a long tail to the right?have a long tail to the left?be symmetric?have the largest value of σ?
46 2.46 Many studies investigating extrasensory perception (ESP) have been conducted. A typical ESP study is carried out by subjecting an individual claiming to have ESP to a series of trials and recording the number of correct identifications made by the subject. Furthermore, when a subject is strictly guessing on each trial, the number of correct identifications can be modeled with a binomial probability model with the probability of a correct identification being p = 0.5 on each trial. If a subject is guessing on each of 20 trials in an ESP study, determinethe probability of 20 correct identifications.the probability of 18 correct identifications.the probability of at least 18 correct identifications.the mean number of correct identifications.
47 2.47 Suppose an individual actually does have ESP and makes correct identifications with probability p = 0.95. If the individual is subjected to a series of 20 independent trials, determinethe probability of making 20 correct identifications.the probability of making fewer than 19 correct identifications.the mean number of correct identifications.
48 2.48 Past studies have shown that 60% of the children of parents who both smoke cigarettes will also end up smoking cigarettes, and only 20% of children whose parents do not smoke cigarettes will end up smoking cigarettes. In a family with four children, use the binomial probability model to determinethe probability that none of the children become smokers given that both parents are smokers.the probability that none of the children become smokers given that none of the parents are smokers.
49 2.49 In Exercise 2.48, is it reasonable to assume that each of the four children will or will not become a smoker independently of the other children? Explain.
50 2.50 Side effects are often encountered by patients receiving a placebo in a clinical trial. Suppose 10 individuals were randomly and independently selected for the placebo group in a clinical trial. From past studies, it is known that the percentage of individuals experiencing significant side effects after receiving the placebo is about 10%. Using the binomial probability model, determinethe probability that two of the 10 patients in the placebo group experience significant side effects.the probability that none of the 10 patients in the placebo group experience significant side effects.the expected number of the 10 patients in the placebo group that will experience significant side effects.the standard deviation of the number of the 10 patients in the placebo group that will experience significant side effects.
51 2.51 If Z has a standard normal distribution, determineP(Z≤−0.76).P(Z<1.28).P(Z≤−2.04).P(Z>0.42).P(Z≥−1.65).P(Z>2.87).P(−1.12<Z≤2.25).P(1.10<Z<2.25).P(−0.80≤Z≤1.22).P(−1.76<Z<−1.26).
52 2.52 If Z has a standard normal distribution, determinethe 5th percentile.the 25th percentile.the 75th percentile.the 98th percentile.the interquartile range.
53 2.53 Intelligence quotient scores are known to follow a normal distribution with mean 100 and standard deviation 15. Using the normal probability model, determinethe probability that an individual has an IQ score