Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 78

1 2.1 What is the difference between a qualitative and a quantitative variable?

2 2.2 What is the difference between a discrete and a continuous variable?

3 2.3 What is the difference between a nominal and an ordinal variable?

4 2.4 Determine whether each of the following variables is a qualitative or quantitative variable.AgeSystolic blood pressureRaceGenderPain level

5 2.5 Determine whether each of the following variables is a qualitative or quantitative variable. If the variable is a quantitative variable, determine whether it is a discrete or a continuous variable. If the variable is a qualitative variable, determine whether it is a nominal or an ordinal variable.GenderEthnicityDosage of a drug given in whole mgAbdomen circumferenceWhite blood countBody mass index (BMI)Eye colorSurvival time after diagnosis with pancreatic cancerNumber of months since last check upNumber of sexual partners in the last 6 months

6 2.6 Determine whether each of the following qualitative variables is a nominal or an ordinal variable. The values that the variable can take on are listed in parentheses following the name of the variable.Gender (M, F)Size of hospital (small, average, large)Blood type (A, B, AB, O)Radiation dosage (low, medium, high)Use of dietary supplements (yes, no)Fat in diet (low, medium, high)Eat lunch (always, sometimes, never)

7 2.7 The percentages given in Table 2.13 were extracted from a bar chart published in the article “Prevalence of overweight among persons aged 2–19 years, by sex—National Health and Nutrition Examination Survey (NHANES), United States, 1999–2000 through 2003–2004” in the November 17, 2006 issue of the Morbidity and Mortality Weekly Report (MMWR), a Centers for Disease Control and Prevention weekly publication.Table 2.13 Prevalence of Overweight Children According to an Article in the November 17, 2006 Issue of MMWRGenderYear1999–20002001–20022003–2004Male1416.518.2Female13.81416Create a side-by-side bar chart representing the percentages of overweight children for each gender by year.Create a side-by-side bar chart representing the percentages of overweight children for each year by gender.

8 2.8 The percentages in Table 2.14 were extracted from a bar chart published in the article “Percentage distribution of blood pressure categories among adults aged ≥18 years, by race/ethnicity—National Health and Nutrition Examination Survey, United States, 1999–2004” published in the June 22, 2007 issue of the Morbidity and Mortality Weekly Report, a Centers for Disease Control and Prevention weekly publication.Table 2.14 Percentages of Americans in the Blood Pressure Categories as Reported in the June 22, 2007 issue of MMWREthnicityBlood Pressure CategoryNormalPrehypertensionHypertensionHypertensionStage IStage IIMexican American4634128White4637116Black36381610Create a side-by-side bar chart representing the percentages for each of the blood pressure categories by ethnicity category.Create a side-by-side bar chart representing the percentages for each of the blood pressure categories within each ethnicity category.Which ethnicity appears to have the largest percentage in the hypertension stage I and II categories?

9 2.9 The probability density of a continuous variable is given in Figure 2.32. If the points labeled A,B,C,D, and E represent the mode, mean, median, 25th percentile, and the 75th percentile, determine which of the points is theFigure 2.32 The probability distribution of the continuous variable X.median of this distribution.mode of this distribution.mean of this distribution.value that only 25% of the values in the population exceed.value that 50% of the values in the population exceed.value that 75% of the values in the population are less than.

10 2.10 If the 25th and 75th percentiles of the distribution given in Figure 2.32 are 38 and 92, determine the value of the interquartile range.

11 2.11 Use the distribution given in Figure 2.33 representing the hypothetical distribution for the survival times for stage IV pancreatic cancer patients to answer the following questions.Figure 2.33 The distribution for Exercise 2.11.Does the distribution appear to be multi-modal?How many modes does this distribution have?Is this distribution symmetric, long-tailed left, or long-tailed right?What is the value of the mode for this distribution?If the points A and B represent the mean and median, which of these two points is the mean?

12 2.12 What is the most common reason that a variable will have a bimodal distribution?

13 2.13 What is the prevalence of a disease?

14 2.14 How is a percentile different from a population percentage?

15 2.15 How do the mean and median differ?

16 2.16 When are the mean and median equal?

17 2.17 Is themean sensitive to the extreme values in the population?median sensitive to the extreme values in the population?

18 2.18 Suppose the population of 250 doctors at a public hospital has been classified according to the variables Age and Gender and is summarized in the table below.25–4041–5556–70Male546642Female244123Determine the percentage of doctors at this hospital that are female.Determine the percentage of doctors at this hospital that are aged 56 or older.Determine the percentage of doctors at this hospital that are female and aged 41 or older.Determine the percentages of doctors at this hospital in each age group.Determine the age group that the median age falls in.

19 2.19 Describe how the geometric mean (GM) is computed and why it might be used in place of the arithmetic mean.

20 2.20 What are three parameters that measure thetypical values in a population.the spread of a population.

21 2.21 Which of the parametersmeasuring the typical value in a population are not sensitive to the extreme values in a population?measuring the spread of a population are not sensitive to the extreme values in a population?

22 2.22 According to the article “Mean body weight, height, waist circumference, and body mass index among adults: United States, 1999 – 2000 through 2015 – 2016” published in National Health Statistics Report (Fryar, 2018), the estimated mean weight of an adult male in the United States in 2015 – 2016 was 197.8. Suppose the distribution of weights of adult males is a mound shaped distribution with mean µ = 200 and standard deviation σ = 25. Determinethe weight range that approximately 95% of the adult males in the United States in 2015 – 2016 fall in.the coefficient of variation for the weights of adult males in the United States in 2015 – 2016.

23 2.23 For a mound-shaped distribution what is the approximate percentage of the population fallingbetween the values μ−2σ and μ+2σ.above the value μ+3σ.below the value μ−σ.

24 2.24 Which parameter measures the relative spread in a population? How is this parameter computed?

25 2.25 What does it mean whenthe median is much larger than the mean?there is a large distance between the 25th and 75th percentiles?there is a large distance between the 75th and 99th percentiles?

26 2.26 In the article “Mean body weight, height, waist circumference, and body mass index among adults: United States, 1999 – 2000 through 2015 – 2016” published in National Health Statistics Report (Fryar, 2018), the statistics in Table 2.15 were reported for adult females in the United States for 2015 – 2016. Use the information in Table 2.15 to answer the following questions.Table 2.15 The Approximate Means and Standard Deviations for the Variables Weight, Height, and Body Mass Index for Adult Females in the U.S. for 2015 – 2016 for Exercise 2.27VariableMeanStandard DeviationWeight169.8 lbs20 lbsHeight63.6 inches3 inchesBMI29.64Compute the coefficient of variation for the variable weight.Compute the coefficient of variation for the variable height.Compute the coefficient of variation for the variable BMI.

27 2.27 What does it mean when the value of the correlation coefficient for two quantitative variables isρ=−1.ρ = 0.ρ = 1.

28 2.28 What does the correlation coefficient measure?

29 2.29 What are the units of the correlation coefficient?

30 2.30 What does it mean when two events are said to beindependent events?dependent events?

31 2.31 Under what conditions is the probability of the event “A or B” equal to the sum of their respective probabilities?

32 2.32 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 of greater than 140.the probability that an individual has an IQ score of less than 80.the probability that an individual has an IQ score of between 105 and 125.the 95th percentile of IQ scores.

54 2.54 Suppose the birth weight of a full-term baby born in the United States follows a normal distribution with mean 7.5 pounds and standard deviation 0.5 pounds. Determine theprobability that a full-term baby born in the United States weighs between 7 and 8 pounds.probability that a full-term baby born in the United States weighs more than 9 pounds.probability that a full-term baby born in the United States weighs less than 6.8 pounds.5th percentile of the weights of full-term babies born in the United States.95th percentile of the weights of full-term babies born in the United States.

55 2.55 According to the National Health Statistics Report Number 122, December 20, 2018 (Fryar, 2018), the estimated mean weight of an adult male in the United States is 197.8 pounds. Suppose the distribution of weights of adult males in the US is normally distributed with mean weight µ = 200 pounds with standard deviation of σ = 25 pounds. Determine theprobability that an adult male in the US weighs more than 240 pounds.probability that an adult male in the US weighs less than 140 pounds.probability that an adult male in the US weighs between 180 and 220 pounds.90th percentile of the weights of adult males in the US.

56 2.56 According to the National Health Statistics Report Number 122, December 20, 2018 (Fryar, 2018), the estimated mean body mass index (BMI) of an adult female in the United States is 29.6. Suppose the distribution of BMI values for adult females in the US is normally distributed with mean BMI µ = 30 with standard deviation of σ = 4. Determine theprobability that an adult female in the US has BMI less than 25.probability that an adult female in the US has BMI more than 36.probability that an adult female in the US has BMI between 26 and 32.10th percentile of the BMI values of adult females in the US.

57 2.57 What are the units of a z score?

58 2.58 How many standard deviations below the mean does a z score of −3 correspond to?

59 2.59 Under what conditions is it possible to determine the percentile associated with an observed z score?

60 2.60 Table 2.18 contains the standard weight classifications based on body mass index (BMI) values. Assuming that BMI is approximately normally distributed, determine the z score corresponding to the cutoff for theTable 2.18 The Standard Weight Classifications Based on BMI ScoresWeight ClassificationBMI Percentile RangeUnderweightLess than 5th percentileHealthy weightBetween 5th and 85th percentilesAt risk of overweightBetween 85th and 95th percentilesOverweightGreater than the 95th percentileunderweight classification.healthy classification.at-risk-of-overweight classification.overweight classification.

61 2.61 Because a BMI value for a child depends on age and sex of the child, z scores are often used to compare children of different ages or sexes. Table 2.19 gives the mean and standard deviation for the distribution of BMI values for male children aged 10 and 15. Use the information in Table 2.19 to answer the following questions concerning two male children, a 10 and a 15 years old, each having a BMI value of 25:Table 2.19 The Mean and Standard Deviations of BMI for 10- and 15-year-old Male ChildrenBMIAgeMeanSD1016.62.31519.83.1Compute the z score for the 10-year-old.Compute the z score for the 15-year-old.Which child has a larger BMI value relative to the population of males in their age group?

62 2.62 According to the National Health Statistics Report Number 122, December 20, 2018 (Fryar, 2018), the estimated mean height of an adult male in the United States is 69 inches and the mean height of an adult female in the United States is 63.6 inches. Suppose the standard deviation of the heights of adult males in the US is σm=3.5 and the adult females in the US is σf=2.5. Determinethe z score associated with an adult male in the US who is 74 inches tall.the z score associated with an adult female in the US who is 70 inches tall.the probability that an adult male in the US is taller than 74 inches. Assume the weights are normally distributed.whether the 74 inch male or the 70 inch female is farther above the mean of their respective population.

63 2.63 In which study designs can therelative risk be computed?odds ratio be computed

64 2.64 In a prospective study on melanoma and the risk factor fair complexion, the probability that an individual will have a melanoma in their life given they have a fair complexion is 0.12, and the probability that an individual will have a melanoma in their life given they do not have a fair complexion is 0.05.Compute the relative risk of developing a melanoma for the risk factor fair complexion.Interpret the relative risk.Compute the odds ratio for developing a melanoma for the risk factor fair complexion.Interpret the odds ratio.

65 2.65 In a prospective study on gum disease, group of 50 subjects will receive an oral wash treatment and a control group of 50 subjects will receive a placebo wash. Suppose four subjects in the treatment group developed gum disease and 16 in the control group developed gum disease.Determine the relative risk of developing gum disease for an individual who receives the treatment.Interpret the relative risk.

66 2.66 In a retrospective study of the health problems associated with smoking, a researcher is interested in the relationship between chronic obstructive pulmonary diseases (COPD) and the risk factor smokes. Use the probabilities in Table 2.66 to answer the following questions.COPDRisk FactorYesNoSmokes0.140.86Does not Smoke0.040.96Compute the odds of having COPD for smokers.Compute the odds of having COPD for non-smokers.Compute the odds ratio for COPD and the risk factor smokes.Interpret the odds ratio.Why is it inappropriate to compute the relative risk in this study?