Biostatistics for the Biological and Health Sciences with Statdisk By Marc M. Triola
Email: Richard@qwconsultancy.com
Contents 1 Introduction
1
2 Describing, Exploring, and Comparing Data
7
3 Probability
63
4 Discrete Probability Distributions
89
5 Normal Probability Distributions
109
6 Estimates and Sample Sizes with One Sample
167
7 Hypothesis Testing with One Sample
201
8 Inferences from Two Samples
235
9 Correlation and Regression
277
10 Multinomial Experiments and Contingency Tables
311
11 Analysis of Variance
343
12 Nonparametric Statistics
359
13 Life Tables
417
Chapter 1: Introduction
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Chapter 1. Introduction 1-2 Types of Data In Exercises 1 – 4, determine whether the given value is a statistic or a parameter. 1. Statistic, since 12 males (the sample size of males) is a characteristic of a sample. 2. Statistic, since 3.8 (the sample mean) is a characteristic of a sample of Canadian geese. 3. Statistic, since 226 m (the sample mean) is a characteristic of a sample of frigate birds. 4. Parameter, since 2.7 years (the population mean) is a characteristic of the population of cloned sheep. In Exercises 5 – 8, determine whether the given values are from a discrete or continuous data set. 5. Discrete, since the number of birds must be in the form of whole numbers, a fraction of a bird is not possible. 6. Continuous, since length in inches of hummingbirds can take on any value over a continuous span. 7. Discrete, since the number of families having guns in their homes must be in the form of whole numbers; a fraction of a home is not possible. 8. Continuous, since head width in inches can take on any value over a continuous span. In Exercises 9 – 16, determine which of the four levels of measurement (nominal, ordinal, interval ratio) is most appropriate. 9. Ratio, since there are equal intervals and a natural starting point of 0 inches. 10. Ordinal, since racing ability can be ordered low to high, but the actual difference in ability between the observed points cannot be determined. 11. Interval, since body temperatures, either measured on a Fahrenheit or Centigrade scale, have equal intervals, but do not have a natural starting point of 0. 12. Nominal, since case numbers would be like labels, they identify but have no ordered property. If the case numbers were in sequence of when they happened over time, they might have an ordinal property. 13. Nominal, since the measure is used as a label for a category and the category does not have any ordinal property. 14. Nominal, since social security numbers are labels with not orderable property. 15. Ratio, since the number of manatees killed by boats has a natural 0 starting point. 16. Nominal, since zip codes are like labels and they have no ordered property. There is a systematic way of assigning them, but that does not provide a way to order them in any way that would be meaningful. In Exercises 17 – 20, identify the (a) sample and (b) population. Also determine whether the sample is likely to be representative of the population. 17. This would be a sample that is likely not representative of the population of rainbow trout, but it could be representative of rainbow trout in the stream where she is netting the trout.
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Chapter 1: Introduction
18. This would be a sample that is likely not representative of the population of persons with attitudes or perceptions about health, but it might be representative of those who read the Florida newspaper. 19. This would be a sample that would likely be representative of a population from which the sample was drawn. 20. This would be a sample that is likely not representative of adult Americans, but it might be representative of the persons the graduate student knows.
1-3 Design of Experiments In Exercises 1 – 4, determine whether the given description corresponds to an observational study or an experiment. 1. Drug Testing Experiment, since the patients are being treated or exposed to a treatment, in this case Lipitor. 2. Treating Syphilis Observation, since patients are observed to determine health condition, but not treated in such a way to modify them. However, this could be part of an experiment if they are compared on the same outcome variables with a treated group. 3. Quality Control Observation, since the amount of aspirin is observed in each tablet, they are not exposed to any treatment. 4. Magnetic Bracelets Experiment, since the use of the bracelets is a treatment that may result in lower likelihood of suffering from the effects of motion sickness. In Exercises 5 – 8, identify the type of observational study (cross-sectional, retrospective, prospective). 5. Medical Research Retrospective, since the study focus is on what has happened in a period of the past (five years) for head injury patients. 6. Psychology of Trauma Prospective, since the study focus is on what happens to siblings of victims of the World Trade Center attacks in a period after the attacks occurred. 7. Flu Incidence Cross-sectional, since the study focus is on what happened at a specific time period. 8. Deer Encounters Retrospective, since the study focus is on what happened relative to damage caused by vehicles hitting deer in a period of the past (10 years). In Exercises 9 – 20, identify which of these types of sampling is used: random, systematic, convenience, stratified, or cluster. 9. Aspirin Usage Convenience, since the researcher asks those who enter the clinic for treatment. 10. Health Survey Selection Systematic, since every ith, in this case every thousandth, name/person in a list is selected for the sample. 11. Telephone Polls Random, since telephone survey respondents were selected at random from a list of phone numbers; each phone number would have the same chance of being selected. However, we need to exercise some caution in calling this random. When a large majority of potential survey respondents had phones and researchers had access to the phone numbers, these types of surveys were often considered random. However, with changes like “no call lists”, unlisted numbers, phones that can be programmed to screen out unrecognized calls, and individuals who have made cell phones their phone of choice and have dropped land line phones, random dialing of phone numbers is less likely to provide a representative sample than has been the case in the past.
Chapter 1: Introduction
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12. Diet Stratified, since categories or strata are identified (four age strata) and 200 study participants are randomly selected from within each strata. 13. Student Drinking Cluster, since classes form the clusters and clusters are randomly selected and all members of the selected clusters become study participants. 14. Clinical Trial of Blood Treatment Random, since each of the treated subjects has the same chance of being selected for the sample that will receive the increased dosage. 15. Sobriety Checkpoint Systematic, since every ith, in this case every fifth, driver is stopped for a sobriety check sample. 16. Orange You Sweet Cluster, since each tree represents a cluster and 20 clusters are randomly selected and all the oranges for the cluster are measured for sugar content. 17. Education and Health Stratified, since categories or strata are identified (three education level strata) and 150 survey participants are randomly selected from within each strata. 18. Anthropometrics Convenience, since family members are selected since they just happen to be around to be interviewed and they are not selected from any larger group. 19. Medical Research Cluster, since each hospital represents a cluster and 30 clusters are randomly selected and all the cardiac patients in the cluster are surveyed. 20. AIDS Survey Stratified, since categories or strata are identified (four marital status strata) and 500 survey respondents are randomly selected from within each strata. Exercises 21 – 26 relate to random samples and simple random samples. 21. Sampling Aspirin Tablets Simple random sample, since every possible sample of size 50 has the same chance of being chosen. 22. Sampling Students Random sample, since each student has the same chance of being selected, but every possible sample of five students (the number in each row) does not have the same chance of being selected for the sample. 23. Convenience Sample Neither a random nor simple random sample, since random selection is not done in any way to select the sample. 24. Systematic Sample Random, since each thermometer has the same chance of being selected, but not every possible sample has the same chance of being selected since the way this is done, the thermometers between those selected cannot be in included in a different sample. 25. Stratified Sample Simple random sample, since we would assume that within each group, there is a random selection of possible survey respondents whereby every possible sample of size 100 men and 100 women has an equal chance of being selected. 26. Type of Sampling Stratified, since there is an equal across group balance of sex and race participants. This would not have been very likely, although possible, to have achieved this balance totally at random if a random sampling strategy had been used. 27. Cluster Sample Random sample, since the blocks are randomly selected in such a way that all different combinations of 10 blocks would have an equal chance of being selected so relative to blocks this would be a simple random sample. However since the blocks are made up of subjects who must remain within the blocks, there is no equal chance for any of the possible samples of all the residents to be selected.