STAT 100 Exam 2 Solutions (75 points) Spring 2016

Transcription

1 STAT 100 Exam 2 Solutions (75 points) Spring In the 1970s, the U.S. government sued a particular school district on the grounds that the district had discriminated against black persons in its hiring of school teachers. The statistical evidence introduced was similar to the following: of the 300 teachers hired in this district in 1972 and 1973 (the years following the passage of the Civil Rights Act), only 11 were black (note that 11/300 = 3.67%). However, about 10% of teachers hired in neighboring school districts (which were similar to this district) that year were black. You have been asked to carry out a simulation study with 1,000 simulated runs in Tinkerplots to determine what outcomes tend to occur by chance if the school district was NOT discriminating based on race in its hiring of teachers. Answer the following questions based on this scenario. a. Fill in the three boxes on the spinner below to set up this simulation. Box 1 is the Repeat value; Box 2 is the probability the hired teacher is black; Box 3 is the probability the hired teacher is not black. (4 pts) b. How many (give a number, not a percentage) of the 300 teachers hired do we expect to be black if there is NO discrimination in the hiring process? (2 pts) 10% of 300 = 30 1

2 The following outcomes from 1,000 runs of the simulation were obtained assuming this district was NOT discriminating based on race in the hiring process. Each dot represents the number out of those hired that were black for each run of the simulation. Answer the following / questions. (2 pts each) a. The above plot shows that in exactly 28 of the 1,000 runs of the simulation study, 71 of the 300 teachers hired were black. b. We would have statistical evidence indicating that the black teachers were discriminated against in this district if 30 of the 300 teachers hired had been black because this result occurred most often in our simulation study. c. Using the 5% rule and the observed result from the study (11 of the 300 teachers hired were black), there is enough statistical evidence to say there was racial discrimination against blacks in the hiring of teachers in this district. d. Suppose this school district had instead hired 20 black teachers (out of the 300 total hired). Using the 5% rule, an observed result of 20 would not provide enough statistical evidence to say there was racial discrimination against blacks in the hiring of teachers in this district. 2

3 2. Consider the example from Handout 12 regarding an investigation of a person who claimed he had suffered hearing loss. Recall that in this experiment, a sound was played with one of two possible lights (red or blue). The subject was asked to correctly identify the light associated with the sound being played. The individual under investigation was thought to be lying about his inability to hear, so the experimenter asked the subject to complete 100 trials. He got only 36 correct. In an effort to evaluate whether or not this would be an unusual outcome for a deaf person, the experimenter obtained outcomes from 100 simulated experiments in which the subject was known to have been guessing (recall how we did this in class by muting the sound). In each of the 100 simulated experiments, the subject completed 100 trials, and these simulated outcomes (i.e., the number they got correct out of the 100 trials) are plotted on the bottom number line below. Subject s Outcome Likely outcomes obtained if the subject is truly deaf Answer the following true/false questions by circling the correct response. (2 pts each) a. In the simulation that produced the results shown above, the Repeat value was set to 36 because the subject got 36 correct. b. Each dot on the above plot represents whether a subject was correct or incorrect when asked which light was displayed on a single trial of the simulated experiment. c. The subject s outcome (36 correct) and the results of the simulation study shown above indicate that we do have statistical evidence that the subject is intentionally giving wrong answers on purpose. d. It is likely that the subject would have answered correctly on only 36 of the 100 trials if he was truly deaf and therefore guessing each time. e. If the experiment had consisted of asking the subject which color of light was displayed only 25 times instead of 100 (so that there were 25 trials in the actual study), then the Repeat value would be set to 25 in his Tinkerplots simulation. f. Suppose the experimenter decided to obtain 1,000 simulated experiments (so that 1,000 dots would appear on the bottom graph), instead. If he still wanted to simulate the experiment described above which consisted of 100 trials, he would change the Repeat value from 100 to 1,000 in his Tinkerplots simulation in order to get 1,000 simulated results. g. Suppose this experiment had asked the subject to choose between three lights instead of two (say red, blue, and green). To simulate outcomes expected when guessing, the spinner would be set up as 50% correct/ 50% incorrect because there would be only two possible outcomes: (1) the subject guessed correctly, or (2) the subject guessed incorrectly. 3

4 3. Researchers often use alternative forced choice procedures to assist in the evaluation of subjects they suspect are exaggerating their health issues. In one such case, a patient claimed to suffer from memory loss so severe that they couldn t remember what had happened only seconds earlier. To investigate this claim, researchers presented the subject with one of four objects: a black pen, a green highlighter, a yellow pencil, or an orange marker. After presentation of the object, the researcher waited 10 seconds and then asked the subject to recall which of these four objects had been displayed (note: they weren t allowed to say, I don t know they were forced to answer with one of the four options). This process was repeated a total of 30 times. To determine what outcomes we expect to see when a subject is guessing, we will carry out a simulation study in Tinkerplots. Answer the following questions regarding the set-up of the simulation study. a. In each simulated experiment, a trial will consist of the subject identifying the object that was shown. How many trials must we simulate (i.e., what is the Repeat value)? (2 pts) 30 b. Each trial will result in either a correct or an incorrect answer. What is the chance, or probability, of each outcome occurring if the subject truly suffers from memory loss and is therefore guessing? (2 pts) Probability of correct answer = 25% Probability of incorrect answer = 75% c. The following graph shows 1,000 simulated outcomes to help you understand what outcomes occur by chance when the subject is truly guessing. Suppose that in the actual study, the subject got only 4 out of 30 trials correct. Based on the results of the simulation study and the 5% rule, does this outcome provide statistical evidence that the subject is answering wrong on purpose? Explain your reasoning. (3 pts) No. An observed outcome of 4 is not in the bottom 5% of this distribution, so it is not low enough to provide statistical evidence the subject is answering wrong on purpose. 4

5 Questions 4 and 5 concern the following scenario. A designed experiment was carried out by obtaining a random sample of subjects with high blood pressure and then randomly assigning each participant either to walk for half an hour three times a week or to sit quietly reading a book for half an hour three times a week. At the end of a year the change in participants' blood pressure over that year was measured, and the change was compared for the two groups. 4. This is a designed experiment rather than an observational study because (2 pts) a. Blood pressure was measured at the beginning and end of the study. b. The two groups were compared at the end of the study. c. Participants were randomly selected for this study. d. The participants were randomly assigned to either walk or read. 5. If at the end of the study it was found that the walkers saw a significantly bigger drop in blood pressure than did the readers, then which of the following is the most correct conclusion? (2 pts) a. It can be concluded that walking vs. reading caused a bigger drop in blood pressure because of the way the study was done; that is, the randomization of participants to treatment groups would have balanced out the effects of any possible lurking variables. b. It can be concluded that walking vs. reading caused a bigger drop in blood pressure because the subjects were randomly selected to participate in this study. c. It cannot be concluded that walking vs. reading caused a bigger drop in blood pressure because in the course of a year there are lots of possible lurking variables that might also affect this. d. It cannot be concluded that walking vs. reading caused a bigger drop in blood pressure because it might be that people with high blood pressure were more likely to read a book than to walk. Answer the following true/false questions. (2 pts each) 6. Researchers want to investigate whether studying Latin helps students achieve higher scores on the verbal section of the SAT exam. Suppose they obtained a random sample of students who had taken the SAT exam last year, some of whom had taken at least 1 year of Latin prior to the SAT exam (note that students were not randomly assigned to study Latin or not they chose this on their own). This is an example of a designed experiment. 7. Consider the previous problem. When comparing records of these students, it is noted that the average SAT verbal score is higher for those students who have taken Latin. Therefore, researchers can conclude there is a causal relationship between studying Latin and achieving higher scores on the verbal section of the SAT exam. 8. In general, if a relative risk ratio is equal to 1, this means there is no significant difference in the outcome of interest between the two groups. 9. A study investigating whether a relationship exists between a student s gender and whether they skip class at least once a week was conducted. Suppose the odds ratio associated with skipping class for males compared to females was 2.11, and the relative risk of skipping class for males compared to females was 1.58 (the risk and odds of skipping class for males were used in the numerator of the risk ratio and odds ratio, respectively). This means that the males in this sample are over twice as likely to skip class at least once a week as are the females. 5

6 10. Polygraph tests (lie-detector tests) are often routinely administered to employees or prospective employees in sensitive positions. Suppose that only 2% of the employees who take the polygraph actually lie. Furthermore, studies conducted to investigate the reliability of polygraph tests have revealed the following. The probability of a positive test result given the subject is actually lying is 88%. The probability of a negative test result given the subject is telling the truth is 86%. a. Consider a hypothetical population of 10,000 employees subjected to such a polygraph test. Use the above information to fill in the following table. (4 pts) Polygraph Test Positive Polygraph Test Negative Total Subject is lying Subject is telling the truth Total ,000 b. Use the counts in the above table to find the positive predictive value of the polygraph test. In other words, find the probability an employee is actually lying given he/she has a positive test result on the polygraph. Show your work in order to obtain full credit. (2 pts) 176/1548 =.11 = 11% c. Use the counts in the above table to find the negative predictive value of the polygraph test. In other words, find the probability an employee is actually telling the truth given he/she has a negative test result on the polygraph test. Show your work in order to obtain full credit. (2 pts) 8428/8452 =.997 = 99.7% 6

7 11. A Story from Bayport, MN On May 22, 2001, the Washington County Department of Public Health and Environment received reports from physicians of children from an elementary school in Bayport, MN, with acute Salmonella infections. A Science Club instructor in the school district had students dissect owl pellets on the tables in the cafeteria. Students dissecting the pellets were not instructed to wash their hands afterwards. Furthermore, the tables were not washed following the dissections, and later that same afternoon kids in the Adventure Club (an after-school program) had their afternoon snack on the same tables. In an effort to determine the source of the Salmonella outbreak, stool samples from the 137 children at Bayport Elementary who exhibited some symptoms of Salmonella infection were collected and analyzed to see if Salmonella was present. Consider the following results which are tabulated according to whether the child was in the Science Club. Science Club Salmonella Yes No Totals Yes No Totals a. The 100% stacked bar chart displays the row percentages for this problem. Show the work behind these percentages. In other words, find the risk of contracting Salmonella for each group. (4 pts) Risk of Salmonella for those in the Science Club = 22/25 = Risk of Salmonella for those not in the Science Club = 17/112 = b. Find and interpret the risk ratio. (3 pts).8800/.1518 = 5.8 (the students in the Science Club were 5.8 times as likely to have salmonella as were those not in the Science Club) 7

8 c. Now, find the odds of Salmonella for each group. (4 pts) Odds of Salmonella for those in the Science Club = 22/3 Odds of Salmonella for those not in the Science Club = 17/95 d. Find and interpret the odds ratio. (3 pts) (22/3) / (17/95) = (the odds of salmonella were 41 times greater for students in the Science Club as for those students not in the Science Club) e. Recall that students in the Adventure Club ate their afternoon snacks on the same tables after the owl pellets were dissected. Suppose that students in the Adventure Club were also investigated and a similar 100% stacked bar chart was constructed. Also, suppose that the following risk ratio was computed. Risk Ratio = Risk of Salmonella for those in the Adventure Club Risk of Salmonella for those NOT in the Adventure Club Shade the Adventure Club = No column to show how the graph would appear if the risk ratio for the Adventure Club was found to be Risk Ratio = 2.0. (2 pts) 8

9 12. Consider a study in which the risk ratios associated with various risk factors for persistent post-partum depression were being considered (persistent means that symptoms lasted up to one year after giving birth). Source: Rahman and Creed. Outcome of prenatal depression and risk factors associated with persistence in the first postnatal year: Prospective study from Rawalpindi, Pakistan. Journal of Affective Disorders June, 100 (1-3): Risk factor Risk Ratio (i.e., Relative Risk) Financially Empowered.8 20 years of age.9 Pregnant for the First Time.6 Husband Not Educated 1.7 Not Educated 1.2 Poverty 1.4 For example, the first risk ratio was computed as follows: Risk Ratio = Risk of Persistent Post - Partum Depression for Women who are Financially Empowered Risk of Persistent Post - Partum Depression for Women who are NOT Financially Empowered a. Of the risk factors listed, which appears to be the least important in determining whether or not a woman will experience persistent post-partum depression? Explain your reasoning. (2 pts) Less than or equal to 20 years of age. This risk ratio was farthest from one, indicating the biggest difference in risk of post-partum depression between groups. b. Of the risk factors listed, which appears to be the most important in determining whether or not a woman will experience persistent post-partum depression? Explain your reasoning. (2 pts) Husband not educated. This risk ratio was farthest from one, indicating the biggest difference in risk of post-partum depression between groups. 9