Final Exam - MATH 105 (Fall 2016) Question # 1 (7 pts) 2 (10 pts) 3 (8 pts) 4 (7 pts) 5 (8 pts) 6 (10 pts) 7 (8 pts) 8 (10 pts) 9 (8 pts) 10 (8 pts) 11 (6 pts) 12 (10 pts) TOTAL: ____________________________________ Name Question Score (7 points): Calculate the statistical measures. If a measure does not exist, please indicate. [7 points] Raw data: [-4, 0, 2, -6, 8, 2, 0, 6, 1] Ordered from least to greatest: Mean:_____________________ Median:___________________ Mode:_____________________ Range:_____________________ A sample from a large population: [10, 20, 0] Use the following formula to find the sample's standard deviation: = ( Question 1 ) Mean: _____________________ Sample standard deviation:___________ = (10 points): Order the 20 data values from least to greatest and create a stem-and-leaf chart with a class/bin size of 10, starting from 0. Raw data: 12 82 28 93 25 52 51 32 77 37 Ordered data: [2 points] Page 1 of 6 42 58 66 33 41 64 48 49 72 45 Stem-and-leaf display: [4 points] 3 0 1 2 3 4 5 6 7 8 9 8 Create the histogram for the above data set with a class/bin size of 10, starting from 0. [4 points] Represents the number 38 Question (8 points): Which data has the smallest standard deviation? Please circle A, B, or C. You do not need to calculate the actual value. [4 points] A: [270, 271, 269, 271, 269] B: [-270, 271, -269, 270, 269] C: [270, 0, 0, 0, -270] Consider the data set with the smallest standard deviation. What would happen to the standard deviation if we'd replace the median value by 0? Justify your reasoning by circling the correct answers: [4 points] The standard deviation would increase / decrease because the new data point (zero) would create a larger/smaller spread in the data. Question (7 points): Assuming normally distributed data, please fill out the empty boxes. (Hint: recall the empirical rule about normal data distribution.) [7 points] Enter the values in the seven boxes under the bell curve so that they correspond to a distribution with arithmetic mean 200 and standard deviation 15. Question (8 points): For which regression line is the coefficient of determination (explaining power) the greatest? [2 points] A B C D In which case is the response variable most sensitive to changes in the explanatory variable? [2 points] A B C D Page 2 of 6 In which case would you say that there is no correlation between the two variables? [2 points] A B C D Which regression line can potentially show the relationship between the dosage of an effective blood pressure-lowering medication (x-axis) and a patient's systolic blood pressure (y-axis)? [2 points] A B Question C D (10 points): Find the interquartile range (Q3-Q1) for the following data: [5 points] 10 20 0 -10 20 15 25 30 30 10 -10 0 -20 Order the data from smallest to largest: Median = Q2= ____________ First quartile = Q1 = ________ Third quartile = Q3 = _______ Smallest value = ____________ Largest value = ____________ Interquartile range = ___________ Which box-and-whiskers plot shows the above dataset? Circle the letter. [5 points] Question (8 points): Which linear model best describes the following data? X 1 2 3 4 5 6 7 Y 7 8 6 4 5 4 2 Plot these data points on EACH graph below. [3 points] Then draw each line (a, b, and c) and select the best-fitting line. [3 points] Page 3 of 6 8 0 9 1 0 15 (a) = 2 + 0.5 (b) (c) =9 Answer: Regression line _______ seems to best fit the data points. [2 points] Question = 1+2 (10 points): Each of these three students in different sections of a math class scored 10 points higher than their respective class average. You can assume that the scores are normally distributed. Fill out the table below: calculate the z-scores for each student [5 points] and rank each student's performance relative to his/her section. [5 points] Score Class average Class standard deviation z-score Rank (1-best, 2-middle, 3-worst) Alex 70 60 15 Bella 85 75 10 Charlie 65 55 5 Put your calculations here: Question (8 points): Samples are taken from two normal populations. The standard deviations of the populations are known. In which case do you get a smaller spread for the estimate of the population mean (in other words a more accurate estimate)? [Hint: recall that the standard deviation of the sampling distribution of the mean depends both on the standard deviation (spread) of the population and the size of your sample.] Sample size: Population standard deviation: Standard error of the mean Rank (1-better estimate, 2-worse estimate) Population A 36 36 Population B 400 100 [4 points] [4 points] Put your calculations here: Page 4 of 6 Question 10 (8 points): A new anemia medication resulted in a 24-point mean increase (posttest minus pretest) in the number of red blood cells in 64 patients. Assume that the distribution of differences is mound-shaped and symmetric. The standard deviation of the sample is 64 and you can assume the same for the population. Can we conclude with a 99% confidence level that the anemia medication is effective? [Hint: calculate the z-score assuming that the null-hypothesis is true and state the probability that our sample (or a more extreme sample) could be drawn assuming that the medication is not effective.] = 24 = 64 Specify your null-hypothesis: Specify your alternate hypothesis: : : < > =0 < > =0 (circle one operator: <, or > or =) [1 points] (circle one operator: <, or > or =) [1 points] Calculate the z-score when we assume that the null-hypothesis is true: [2 points] = = Find the probability that our sample (or a more extreme sample) could be randomly drawn from the entire population when the null-hypothesis is true. [Hint: Interpret the z-score and use the empirical rule about normal distributions.] [2 points] Conclusion: [2 points] We should reject / accept the null-hypothesis. The medication is / is not effective at the 99% confidence level because the probability of ever seeing a sample as extreme, or more extreme than ours when the medication is not effective is smaller/larger than 1%. Question 11 (6 points): A medical study found that the blood pressure of adults who drink 5 cups of coffee or more per day was higher than the blood pressure of adults who drank no coffee at all. The mean difference was 15 points with a 95% confidence interval of 3 points. The study sampled 100 people in each group. What would be the 95% confidence interval if the study had sampled 10,000 people per group? [3 points] Based on this study alone, can we claim that drinking 5 cups of coffee or more per day has an effect on blood pressure? [3 points] Yes No Page 5 of 6 Question 12 (10 points): A new vaccine provides 80% protection against a certain disease (p=0.8). Five hundred emergency responders are sent into a city that was hit by an outbreak. Calculate the probability that at least 450 of the responders remain protected against the disease. Is this a binomial problem? [2 points] Is there a fixed number of experiments? Yes No Does each trial have only two outcomes? Yes No Are the outcomes of each trial independent? Yes No Is the probability of success the same in each trial? Yes No Based on the above answers, the probability distribution is Binomial Not binomial. Can normal approximation be used to calculate this probability? Calculate the necessary conditions and answer the question below. [2 points] = ____________ Normal approximation can / cannot be used because = _________ is greater / smaller than _______. Calculate the expected value (mean) and the standard deviation of the normal approximation. Show the formulas and your work. [2 points] Formulas: Calculated values: = = ____________ = = _____________ Calculate the probability that at least 450 of the 500 responders remain protected against the disease. Show your work. [4 points] --- END --Page 6 of 6