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Exam Final Solutions - MSM 709 (Data, Statistics, and Probability for Middle School Teachers) - Summer II, 2008 DIRECTIONS. A necessary part of a complete solution is a clear and thorough explanation of how you arrived at your answer. Good luck. 1. (10 points) These questions refer to the histogram pictured below 14 12 10 8 6 4 2 0 5 10 15 20 25 (a) This histogram is A. skewed left B. skewed right C. symmetric D. None of the above Answer: A. skewed left (b) Which of the following is true of the data that produced this histogram A. m > x B. m < x C. m x D. None of the above Answer: A. m > x 2. This question refers to the data on SAT math scores below. The variable SATV is the mean SAT verbal score in the state indicated in 2002. (a) Calculate the five number summary for this data. Answer: min Q1 m Q3 max 1 = = = = = 488 509 542 576 597 (b) Sketch a box plot for the data. Answer: 500 520 540 560 580 (c) Calculate the mean and standard deviation for this data. Answer: x = 541, s = 38.7 (d) Which (if any) of the observations in this data set qualify as outliers using the 1.5IQR rule? Answer: None 2002 SAT data State SATM Mississippi 547 North Dakota 610 Arkansas 556 Iowa 602 South Dakota 586 Utah 559 Wisconsin 599 State Nevada Arizona Alaska California Hawaii Washington Texas SATM 518 523 519 517 520 529 500 3. (5 points) Briefly, what is the difference between an experiment and an observational study? Answer: In an experiment, the experimenter imposes some treatment on the subjects of the experiment (i.e., the individuals in the data set), seeking to analyze the effect (if any) that this treatment will have on whatever variable is being measured. In an observational study, some variable is measured without any intervention on the part of the researcher (i.e., there is no treatment). 4. (5 points) A researcher conducts an opinion poll of Salem residents by randomly selecting 100 phone numbers from the Salem phone book and calling those numbers to ask peoples opinion. What are some issues with regarding this data as a simple random sample from the population of Salem residents? 2 Answer: Some Salem residents dont have phones. Some who have phones arent listed in the phone book. In any event, the researchers sampling technique will result in some undercoverage. Also, non-response will likely be a big factor in the data. 5. An experiment seeks to analyze how people can improve their scores on the LSAT exam (part of the admissions process at most law schools). The study considers four different LSAT preparatory courses call them course A, course B, course C, and course D and assigns some participants in the study to receive some one-on-one tutoring, while the others don't receive any one-on-one tutoring. At the conclusion of the preparatory program, the participants take the LSAT a second time and the changes in their scores are recorded. (a) What are the factors in the experiment? Answer: The preparatory course taking is one factor, and the presence or absence of one-on-one tutoring is the other. (b) How many treatments are there in the experiment? Answer: Eight. (c) What is/are the response variable(s)? Answer: The change in LSAT score. 6. (15 points) Suppose that the mean age of the residents of the city of Imagiton is 46 and that the ages of Imagiton residents are normally distributed with a standard deviation of 12. (a) What fraction of Imagiton residents are more than 60 years old? Answer: 12.17% (b) If I take a simple random sample of 36 Imagiton residents, what is the probability that the mean age in my sample is less than 42? Answer: .02275 (c) What age marks the first quartile of the age distribution of Imagiton residents (i.e., what age marks the top of the lower 25% of the age distribution)? Answer: 37.91 3 7. (15 points) A sample of 11 newborns found that their longest stretches of uninterrupted sleep in a 24-hour period were as follows (measurements in minutes) 119 184 133 101 176 205 114 107 155 168 139 (a) Compute a 90% confidence interval for the mean length of the longest stretch of uninterrupted sleep for a newborn. Answer: (126.69, 164.4) (b) Does this data provide significant evidence that the average length of newborns' longest stretch of uninterrupted sleep is less than two hours forty minutes? Give the hypotheses, test statistic, and pvalue of an appropriate statistical test. Answer: Ha : < 160, H0 : = 160, t = -1.389, p = .097 (c) What assumptions must be made or conditions met in order for your answers to (a) and (b) to be accurate? Answer: The sample must be a simple random sample of all newborns, and the population must be normally distributed. 8. A researcher sampled 43 two-year old boys at random and measured their height and weight. She reported that 31 4.2 inches was a 95% confidence interval for the mean height. Which of the following statements are correct (more than may one be correct) (a) Ninety-five percent of the time, the mean height of two-year old boys is between 26.8 and 35.2 inches. (b) There is a ninety-five percent probability that the interval [26.8, 35.2] captures the true mean height of all two-year old boys. (c) The margin of error in this confidence interval is 4.2. (d) Ninety-five percent of all two-year old boys have a height between 26.8 and 35.2. (e) The mean in the researcher's sample was 4.2 (f) The standard deviation in the researcher's sample was 4.2 (g) The mean in the researcher's sample was 31. 4 Answer: (b), (c), and (g) are accurate. 9. Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with less than 12 grams of hemoglobin per deciliter (g/dl) of blood are anemic. A public health official in Jordan suspects that the mean for all children in Jordan is less than 12. He measures a sample of 25 children, computes an average hemoglobin level in his sample of 10.8 g/dl. Suppose we know that the hemoglobin level for children follows a normal distribution with standard deviation = 2.1. (a) (5 points) Formulate null and alternative hypotheses for a statistical test that will test the official's suspicion. Is this a one-sided or a two-sided test? Answer: Ha : < 12, H0 : = 12; one-sided. (b) (3 points) Calculate the test statistic. Answer: z = -2.857 (c) (3 points) Compute the p-value. Answer: p = .00214 (d) (3 points) Is the data significant at the level of .25? At the level of .05? Answer: Yes and yes. (e) (5 points) If the mean hemoglobin level in the sample had been 11.1 g/dl, would the data be more or less significant (all other things remaining the same)? Answer: Less significant. 10. A researcher designed two computer lessons, one which required children to actively participate in the lesson and another in which children could only affect the pace of the lesson. We'll call the first lesson "active" and the second one "passive." One group of students was selected randomly and given the active lesson, another group was given the passive lesson. After the lesson, each group was given the same test on the content of the lesson. Below are the numbers of correct answers (out of a total of 35) in each group 5 Active 29 28 24 Passive 16 14 17 31 15 15 26 24 17 27 12 (a) Compute a 95% confidence interval for the mean difference in the number correct between the two groups. Answer: 8.715 6.375 = [3.0206, 14.408] (b) What assumption(s) must be made in order for the interval in part (a) to be valid? Answer: The samples must be independent simple random samples from normal populations. (c) Suppose the two lessons were on different content, and that just one group of seven students took first the one lesson, taking a test at its conclusion and then the second lesson, taking a test at its conclusion. Supposing that this procedure generated the same data as above, compute a 95% confidence interval for the mean difference in the number correct between the two groups. Answer: 8.715 8.721 = [-.006, 17.436] (d) Distinguish between the experimental design described in the problem and the one described in part (c). Answer: The problem describes a two-sample design involving two independent random samples; part (c) describes a matched pairs design. (e) Assuming the experimental design was that described in the problem, is there statistical evidence that the active lesson was more effective than the passive lesson? Report the hypotheses, test statistic, and p-value of an appropriate statistical test. Answer: Hypotheses: H0 : 1 - 2 = 0; Ha : 1 - 2 > 0. Test statistic: t = 3.35. P-value: .003. 11. This problem refers to the data below which gives the annual per capital alcohol consumption among men and the number of male cirrhosis deaths per 100,000 men in the same year for 27 Western countries. 6 Alcohol Consumption (X) 13.3 12.2 11.1 11 11 10.8 10.6 10.6 10.6 9.7 9.6 9.4 9.1 8.8 Cirrhosis Deaths (Y ) 27 23.9 41.6 33.1 40.6 104.5 18.8 28.9 28.9 13.2 14.5 14.5 34.8 28.9 Alcohol Consumption (X) 8.8 8.5 8.4 7.9 7.8 7.7 7.7 7.2 7.2 7 6.9 6.7 5.6 Cirrhosis Deaths (Y ) 3.2 50.1 6.2 88 15 7.2 5.1 13.7 13.7 19.1 11.1 15.6 10 (a) Compute r, r2 , and the least-squares regression line. Answer: r = .314, r2 = .099, y = -9.502 + 3.947x (b) Graph the least-squares regression line over a scatterplot of the data. (c) Use the least-squares regression line to estimate the number of cirrhosis deaths per 100,000 men in a Western country in which the men consumed an average of 14.5 liters of alcohol per year. Answer: 47.73 (d) Use the least-squares regression line to estimate by how much the rate of cirrhosis deaths changes if the male per capita alcohol consumption increases by 1.6 liters. Answer: 6.3152 deaths per 100,000 men. (e) Compute the residual for the fifth observation in the data set. Answer: 6.685 7

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