Math 533. Question 1: A national standard requires that public bridge over 200 feet in length must be inspected and rated every 2 years. The rating scale ranges from 0 (poorest rating to 0 (highest rating). A group of engineers used a probabilistic model to forecast the inspection ratings of all major brides in a city. For the year 2020, the engineers forecast that 9% of all major bridges in that city will have rating of 4 or below. Complete parts a and b. a. Use the forecast to find the probability that in a random sample of 10 major bridges in the city, at least 3 will have an inspection rating of 4 or below in 2020. P(x 3 ) = ? (Round to five decimal places as needed) b. Suppose that you actually observe 3 or more of the sample of 10 bridges with inspection rating of 4 or below in 2020. What inference can you make? Why? Question 2: According to a certain golf association, the weight of the golf ball ball shall not be greater than 1.620 ounces (45.93 grams). The diameter of the ball shall not be less than 1.680 inches. The velocity of the ball shall not be greater than 250 feet per second. The golf association periodically checks the specifications of golf balls using random sampling. Three dozen of each kind are sampled, and if more than three do not meet size or velocity requirements, that kind of ball ir removed from the golf association's approved list. Complete parts a and b. a. What assumption must be made in order to use the binomial probability distribution to calculate the probability that a particular kind of golf ball will be removed? A. The experiment consists of n identical trials. The number of outcomes can vary. The probability of success can change. The trials are independent. B. The experiment consists of n identical trials. There are only two possible outcomes on each trial. The probability of success remains the same from trial to trial. The trials are independent. C. The experiment consists of n identical trials. There are only two possible outcomes on each trial. The probability of success can change from trial to trial. The trials are dependent. What information must be known in order to use the binomial probability distribution to calculate the probability that a particular kind of golf ball will be removed? A. The percentage of that kind of golf ball that meets the velocity requirements. B. The percentage of that kind of golf ball that meets all the requirements. C. The percentage of that kind of golf ball that meets the size requirements. D. The percentage of that kind of golf ball that fails to meet both size and velocity requirements. b. Suppose 15% of all balls produced by a particular manufacturer are less than 1.680 inches in diameter, and assume that the number of such balls, x, in a sample of three dozen balls can be adequately characterized by a binomial probability distribution. Find each mean of the binomial distribution. = (Round to the nearest tenth as needed). Find the standard deviation of the binomial distribution. = (Round to the nearest thousandth as needed). Question 3: If x is a binomial random variable, use the binomial probability table to find the probabilities below. a. P(x=3) for n = 10, p = 0.3 b. P(x 6 ) for n = 15, p = 0.5 c. P(x 1 ) for n = 5, p = 0.6 d. P(x 6 ) for n = 15, p = 0.8 e. P(x 14 ) for n = 25, p = 0,8 f. P(x = 3) for n = 20, p = 0.1 a. P(x = 3) = ? (Round to three decimal places as needed) b. P(x 6 ) = ? (Round to three decimal places as needed) c. P(x 1 ) = ? (Round to three decimal places as needed) d. P(x 6 ) = ? (Round to three decimal places as needed) e. P(x 14 ) = ? (Round to three decimal places as needed) f. P(x = 3) =? (Round to three decimal places as needed) Question 4: The chances of a tax return being audited are about 22 in 1,000 of an income is less than $1,000 and 34 in 1,000 if an income is $100,000 or more. Complete parts a through e a . What is the probability that a taxpayer with income less than $100,000 will be audited? With income of $100,000 or more? P(taxpayer with income less than $100,000 is audited) = ? (Type an integer or a decimal) What is the probability that a taxpayer with income of $100,000 or more will be audited? P(taxpayer with income of $100,000 or higher is audited) = ? (Type an integer or a decimal.) b. If three taxpayers with income under $100,000 are randomly selected, what is the probability that exactly one will be audited? That more than one will be audited? P(x=1) = ? (Round to four decimal places as needed) What is the probability that more than one will be audited? P(x 1 ) = ? (Round to four decimal places as needed) c. Repeat part b assuming that three taxpayers with income of $100,000 or more are randomly selected. P(x=1) = ? (Round to four decimal places as needed) What is the probability that more than one will be audited? P(x 1 ) = ? (Round to four decimal places as needed) d. If two taxpayers with incomes under $100,000 are randomly selected and two with incomes more than $100,000 are randomly selected, what is the probability that none of these taxpayers will be audited? P(none of the taxpayers will be audited) = ? (Round to four decimal places as needed) e. What assumption did you have to make in order to answer these questions? A. We must assume that the variables are binomial random variables. We must assume that the trials are identical, the probability of success is the same from trial to trial, and that the trials are independent. B. We must assume that the variables are binomial random variables. We must assume that the trials are identical and dependent. C. We must assume that the variables are random variables. We must assume that the trials are identical, and the probability of success varies from trial to trial. D. We must assume that the variables are binomial random variables. We must assume that the trials are identical, the probability of success varies from trial to trial, and that trials are dependent. Question 5: According to a consumer survey of young adults (18-24 years of age) who shop online, 17% own a mobile phone with internet access. In a random sample of 200 young adults who shop online, let x be the number who own a mobile phone with internet access. a. Explain why x is a binomial random variable (to a reasonable degree of approximation). Choose the correct explanation below. A. The experiment consists of n identical, dependent trials, where there are only two possible outcomes, S (for Success) and F (for Failure). B. The experiment consists of n identical, dependent trials, with more than two possible outcomes. The probability that an event occurs varies from trial to trial. C. The experiment consists of n identical, independent trials, where there are only two possible outcomes, S (for Success) and F (for Failure). The probability of S remains the same from trial to trial. The variable x is the number of S's in n trials. D. The experiment consists of n identical, independent trials, where there are only two possible outcomes, S (for Success) and F(for Failure). The probability of S varies from trial to trial. The variable x is the number of F's in n trials. b. What is the value of p? Interpret this value. p = ? (Type an integer or a decimal) Choose correct interpretation of p below. A. For any young adult, the probability that they own a mobile phone with internet access is 1 - p. B. For any young adult, the probability that they own a mobile phone with internet access is np. C. For any young adult, the probability that they do not own a mobile phone with internet access is p. D. For any young adult, the probability that they own a mobile phone with internet access is p. c. What is expected value of x? Interpret this value. E(x) = ? Choose correct interpretation below. A. In a random sample of 200 young adult E(x) is the average number of young people surveyed that will not own mobile phones with internet access. B. In a random sample of 200 young adults E(x) will always the number of young people surveyed that will own mobile phones with internet access. C. In a random sample of 200 young adults E(x) is the average number of young people surveyed that will own mobile phones with internet access. D. In a random sample of 200 young adults E(x) is the largest possible number of young people surveyed that will own mobile phones with internet access. Question 6: A country's government has devoted considerable funding to missile defense research over the past 20 years. The latest development is the Space-Based Infrared System (SBIRS), which uses satellite imagery to detect and track missiles. The probability that an intruding object (e.g., a missile) will be detected on a flight track by SBIRS is 0.6. Consider a sample of 10 simulated tracks, each with an intruding object. Let x equal the number of these tracks where SBIRS detects the object. Complete parts a through d. a. Give the values of p and n for the binomial distribution. p = ? (Round to one decimal place as needed) n=? b. Find P(x=3), the probability that SBIRS will detect the object on exactly 3 trakcs. P(x=3) = ? (Round to three decimal places as needed) c. Find P(x 3 ), the probability that SBIRS will detect the object on at least 3 tracks. P(x 3 ) = ? (Round to three decimal places as needed) d. Find E(x) and interpret the result. A. E(x) = 4. For every 10 intruding object, SBIRS will detect an average of 4. B. E(x) = 3. For every 10 intruding objects, SBIRS will detect an average of 3. C. E(x) = 6. For every 10 intruding objects, SBIRS will detect an average of 6. D. E(x) = 2.4. For every 10 intruding objects, SBIRS will detect an average of 2.4 E. E(x) = 5. For every 10 intruding objects, SBIRS will detect an average of 5 Question 7: Zoologists investigated the likelihood of fallow deer bucks fighting during the mating season. During a 270-hour observation period, the researchers recorded 205 encounters between two bucks. Of these, 183 involved one buck clearly initiating the encounter with the other. In these 183 initiated encounters, the zoologists kept track of whether or not a physical contact fight occurred and whether the initiator ultimately won or lost the encounter. (The buck that is driven away by the other is considered the losers). Suppose we select one of these 183 encounters and note the outcome (fight status and winner). Initiator Wins No Clear Winner Initiator Loses Totals Fight 28 24 18 70 No Fight 84 Totals 112 15 39 14 32 113 183 a. What is the probability that a fight occurs and the initiator wins? The probability is = ? (Round to the nearest thousandth as needed) b. What is the probability that there is no fight occurs? The probability is = ? (Round to the nearest thousandth as needed) c. What is the probability that there is no clear winner? The probability is = ? (Round to the nearest thousandth as needed) d. What is the probability that a fight occurs or initiator loses? The probability is = ? (Round to the nearest thousandth as needed) e. Are the events \"no clear winner\" and \"initiator loses\" mutually exclusive? A. Yes B. No Question 8: A table classifying sample of 143 patrons of a restaurant according to type of meal and their rating of the service is shown below. Suppose we select, at random, one of the 143 patrons. Give that the meal was lunch, what is the probability that the service was poor? Service good Meals Lunch 24 Dinner 41 Totals 65 Service poor Totals 39 39 78 63 80 143 Given that the meal was lunch, the probability that the service was poor is = ? (Type an integer or a simplified fraction). Course Project: Hospital Stays, Impact of Education, Health Data Introduction In this case, 40 individuals visited the hospital looking for information and education regarding their diabetes. Since diabetes relates to the body's ability to use glucose, their glucose level was measured. These patience took a knowledge test about diabetes before and after being given some educational materials. Satisfaction with their overall visit was also measured through a survey. The data appear below for your reference. The Excel spreadsheet with these data can be found in Doc Sharing and should be opened in Minitab to complete the analyses required in each part of this project. Hosp_Stay Hosp_Satisfaction Diab_Pretest Diab_Posttest Glucose 2 VeryDissat 34 39 122 3 SWDissat 33 30 116 1 SWSat 29 22 108 1 SWSat 17 14 63 7 SWDissat 69 60 74 7 VerySat 69 70 84 8 SWDissat 81 82 67 8 VerySat 81 88 57 9 SWSat 52 100 231 4 SWDissat 45 48 93 1 SWSat 38 28 76 1 VerySat 27 33 163 6 VeryDissat 45 54 217 8 SWDissat 58 71 112 7 SWDissat 68 84 107 6 SWDissat 48 49 95 1 VerySat 9 15 104 8 VerySat 50 51 63 6 SWDissat 65 51 189 6 SWDissat 64 78 53 1 VerySat 25 30 96 5 VerySat 50 45 102 2 VeryDissat 47 41 142 2 VeryDissat 24 18 133 3 VerySat 35 30 88 1 SWDissat 29 31 87 4 SWDissat 39 33 161 4 SWDissat 53 42 104 5 VeryDissat 57 49 92 4 SWSat 50 50 168 8 VeryDissat 64 100 72 6 VeryDissat 90 95 100 8 SWDissat 52 63 63 9 SWSat 52 64 132 9 VerySat 67 77 171 8 SWSat 68 68 163 5 SWSat 67 60 84 7 SWSat 44 53 102 7 VeryDissat 57 51 130 3 SWSat 58 50 71 The variables are 1. Hospital stay - number of days in hospital 2. Hospital satisfaction - level of satisfaction with hospital services during stay (very highly satisfied, highly satisfied, somewhat dissatisfied, and very dissatisfied) 3. Diabetes Pretest - score on diabetes knowledge test (must score 95 or higher to leave) 4. Diabetes Posttest - score on diabetes knowledge test after receiving educational material created for this study 5. Glucose - blood glucose in mg/dL. The data are available in Doc Sharing Course Project Data Set as an Excel file. You are to copy and paste the data set into a Minitab Worksheet. PROJECT PART A: Exploratory Data Analysis . Open the file MATH533_Course_Project_Data_HOSPITAL.xlsx from Doc Sharing. . Summarize the data for each of the five variables. For each variable, find the mean, median, variance, and standard deviation. Use Minitab as appropriate, and explain what the results mean. These calculations may not be possible for each variable. If you cannot calculate any or all of these for a specific variable, please note that and state why . Analyze the connections or relationships between two variables. There are ten pairings possible here (Hospital stay and hospital satisfaction, hospital stay and diabetes pretest, hospital stay and diabetes posttest, hospital stay and glucose, hospital satisfaction and diabetes pretest, hospital satisfaction and diabetes posttest, hospital satisfaction and glucose, diabetes pretest and diabetes posttest, diabetes pretest and glucose, and diabetes posttest and glucose ). Choose two of these pairings and find the correlation coefficient and show the scatter plot. Explain what you see. Some variables show clear relationships, while others do not. . Prepare your report in Microsoft Word (or some other word processing package), integrating your graphs and tables with text explanations and interpretations. Be sure that you have graphical and numerical back up for your explanations and interpretations. Be selective in what you include in the report. You should not generate a 20 page report on every variable and every possible relationship. Rather what you should do is to highlight what you see for the individual and two or three sentences of interpretation. For the two pairings you selected, identify and report your results using graphical and numerical summary (as appropriate), with interpretations. Submission: The report from part 4 including all relevant graphs and numerical analysis along with interpretations. Format for report: A. Brief Introduction B. Discuss your 1st individual variable C. Discuss your 2nd individual variable D. Discuss your 3rd individual variable E. Discuss your 4th individual variable F. Discuss your 5th individual variable G. Discuss your 1st pairing of variables H. Discuss your 2nd pairing of variables I. Conclusion Project Part B: Hypothesis Testing and Confidence Intervals Your manager has speculated the following: a. The average (mean) level of glucose is less than 130 b. The true population proportion of number somewhat satisfied with their hospital stay is greater than 28%, c. the average (mean) score on the diabetes pretest is greater than 52. d. The average (mean) score on the diabetes posttest ONLY among those that are "somewhat satisfied" is less than 75. 1. Using the sample data, perform the hypothesis test for each of the above situations in order to see if there is evidence to support your manager's belief in each case a.-d. In each case use the Seven Elements of a Test of Hypothesis, in Section 6.2 of your textbook with = .05, and explain your conclusion in simple terms. Also be sure to compute the p-value and interpret. 2. Follow this up with computing 95% confidence intervals for Glucose and Diabetes Posttest, and again interpreting these intervals. 3. Write a report to your manager about the results, distilling down the results in a way that would be understandable to someone who does not know statistics. Clear explanations and interpretations are critical. Submission: The report from part 3 + all of the relevant work done in the hypothesis testing (including Minitab) in 1, and the confidence intervals (Minitab) in 2 as an appendix. Format for report: A. Summary Report (about 1 paragraph on each of the speculations a.-d.) B. Appendix with all of the steps in hypothesis testing (the format of the Seven Elements of a Test of Hypothesis, in Section 6.2 of your textbook) for each speculation a.-d. as well as the confidence intervals, and including all Minitab output Project Part C: Regression and Correlation Analysis Using Minitab perform the regression and correlation analysis for the data on diabetes posttest (Y), the dependent variable, and diabetes pretest (X), the independent variable, by answering the following. 1. Generate a scatterplot for diabetes posttest (Y) vs. diabetes pretest (X) including the graph of the "best fit" line. Interpret. 2. Determine the equation of the "best fit" line, which describes the relationship between diabetes posttest and diabetes pretest. 3. Determine the coefficient of correlation. Interpret. 4. Determine the coefficient of determination. Interpret. 5. Test the utility of this regression model (use a two tail test with =.05). Interpret your results, including the p-value. 6. Based on your findings in 1-5, what is your opinion about using diabetes pretest to predict diabetes postest? Explain. 7. Compute the 95% confidence interval for beta-1 (the population slope). Interpret this interval. In an attempt to improve the model, we attempt to do a multiple regression model predicting diabetes posttest based on diabetes pretest and glucose. 8. Using Minitab, run the multiple regression analysis using the variables diabetes pretest and glucose to predict diabetes posttest. State the equation for this multiple regression model. 9. Perform the Global Test for Utility (F-Test). Explain your conclusion. 10. Perform the t-test on each independent variable. Explain your conclusions and clearly state how you should proceed. In particular, which independent variables should we keep and which should be discarded. 11. Is this multiple regression model better than the linear model that we generated in parts 1-7? Explain. Summarize your results from 1-11 in a report that is three pages or less in length and explains and interprets the results in ways that are understandable to someone who does not know statistics. Submission: The summary report + all of the work done in 1-11 (Minitab Output + interpretations) as an appendix. Format: A. Summary Report B. Points 1-11 addressed with appropriate output, graphs and interpretations. Be sure to number each point 1-11. Course Project: Hospital Stays, Impact of Education, Health Data Introduction In this case, 40 individuals visited the hospital looking for information and education regarding their diabetes. Since diabetes relates to the body's ability to use glucose, their glucose level was measured. These patience took a knowledge test about diabetes before and after being given some educational materials. Satisfaction with their overall visit was also measured through a survey. The data appear below for your reference. The Excel spreadsheet with these data can be found in Doc Sharing and should be opened in Minitab to complete the analyses required in each part of this project. Hosp_Stay Hosp_Satisfaction Diab_Pretest Diab_Posttest Glucose 2 VeryDissat 34 39 122 3 SWDissat 33 30 116 1 SWSat 29 22 108 1 SWSat 17 14 63 7 SWDissat 69 60 74 7 VerySat 69 70 84 8 SWDissat 81 82 67 8 VerySat 81 88 57 9 SWSat 52 100 231 4 SWDissat 45 48 93 1 SWSat 38 28 76 1 VerySat 27 33 163 6 VeryDissat 45 54 217 8 SWDissat 58 71 112 7 SWDissat 68 84 107 6 SWDissat 48 49 95 1 VerySat 9 15 104 8 VerySat 50 51 63 6 SWDissat 65 51 189 6 SWDissat 64 78 53 1 VerySat 25 30 96 5 VerySat 50 45 102 2 VeryDissat 47 41 142 2 VeryDissat 24 18 133 3 VerySat 35 30 88 1 SWDissat 29 31 87 4 SWDissat 39 33 161 4 SWDissat 53 42 104 5 VeryDissat 57 49 92 4 SWSat 50 50 168 8 VeryDissat 64 100 72 6 VeryDissat 90 95 100 8 SWDissat 52 63 63 9 SWSat 52 64 132 9 VerySat 67 77 171 8 SWSat 68 68 163 5 SWSat 67 60 84 7 SWSat 44 53 102 7 VeryDissat 57 51 130 3 SWSat 58 50 71 The variables are 1. Hospital stay - number of days in hospital 2. Hospital satisfaction - level of satisfaction with hospital services during stay (very highly satisfied, highly satisfied, somewhat dissatisfied, and very dissatisfied) 3. Diabetes Pretest - score on diabetes knowledge test (must score 95 or higher to leave) 4. Diabetes Posttest - score on diabetes knowledge test after receiving educational material created for this study 5. Glucose - blood glucose in mg/dL. The data are available in Doc Sharing Course Project Data Set as an Excel file. You are to copy and paste the data set into a Minitab Worksheet. PROJECT PART A: Exploratory Data Analysis . Open the file MATH533_Course_Project_Data_HOSPITAL.xlsx from Doc Sharing. . Summarize the data for each of the five variables. For each variable, find the mean, median, variance, and standard deviation. Use Minitab as appropriate, and explain what the results mean. These calculations may not be possible for each variable. If you cannot calculate any or all of these for a specific variable, please note that and state why . Analyze the connections or relationships between two variables. There are ten pairings possible here (Hospital stay and hospital satisfaction, hospital stay and diabetes pretest, hospital stay and diabetes posttest, hospital stay and glucose, hospital satisfaction and diabetes pretest, hospital satisfaction and diabetes posttest, hospital satisfaction and glucose, diabetes pretest and diabetes posttest, diabetes pretest and glucose, and diabetes posttest and glucose ). Choose two of these pairings and find the correlation coefficient and show the scatter plot. Explain what you see. Some variables show clear relationships, while others do not. . Prepare your report in Microsoft Word (or some other word processing package), integrating your graphs and tables with text explanations and interpretations. Be sure that you have graphical and numerical back up for your explanations and interpretations. Be selective in what you include in the report. You should not generate a 20 page report on every variable and every possible relationship. Rather what you should do is to highlight what you see for the individual and two or three sentences of interpretation. For the two pairings you selected, identify and report your results using graphical and numerical summary (as appropriate), with interpretations. Submission: The report from part 4 including all relevant graphs and numerical analysis along with interpretations. Format for report: A. Brief Introduction B. Discuss your 1st individual variable C. Discuss your 2nd individual variable D. Discuss your 3rd individual variable E. Discuss your 4th individual variable F. Discuss your 5th individual variable G. Discuss your 1st pairing of variables H. Discuss your 2nd pairing of variables I. Conclusion Project Part B: Hypothesis Testing and Confidence Intervals Your manager has speculated the following: a. The average (mean) level of glucose is less than 130 b. The true population proportion of number somewhat satisfied with their hospital stay is greater than 28%, c. the average (mean) score on the diabetes pretest is greater than 52. d. The average (mean) score on the diabetes posttest ONLY among those that are "somewhat satisfied" is less than 75. 1. Using the sample data, perform the hypothesis test for each of the above situations in order to see if there is evidence to support your manager's belief in each case a.-d. In each case use the Seven Elements of a Test of Hypothesis, in Section 6.2 of your textbook with = .05, and explain your conclusion in simple terms. Also be sure to compute the p-value and interpret. 2. Follow this up with computing 95% confidence intervals for Glucose and Diabetes Posttest, and again interpreting these intervals. 3. Write a report to your manager about the results, distilling down the results in a way that would be understandable to someone who does not know statistics. Clear explanations and interpretations are critical. Submission: The report from part 3 + all of the relevant work done in the hypothesis testing (including Minitab) in 1, and the confidence intervals (Minitab) in 2 as an appendix. Format for report: A. Summary Report (about 1 paragraph on each of the speculations a.-d.) B. Appendix with all of the steps in hypothesis testing (the format of the Seven Elements of a Test of Hypothesis, in Section 6.2 of your textbook) for each speculation a.-d. as well as the confidence intervals, and including all Minitab output Project Part C: Regression and Correlation Analysis Using Minitab perform the regression and correlation analysis for the data on diabetes posttest (Y), the dependent variable, and diabetes pretest (X), the independent variable, by answering the following. 1. Generate a scatterplot for diabetes posttest (Y) vs. diabetes pretest (X) including the graph of the "best fit" line. Interpret. 2. Determine the equation of the "best fit" line, which describes the relationship between diabetes posttest and diabetes pretest. 3. Determine the coefficient of correlation. Interpret. 4. Determine the coefficient of determination. Interpret. 5. Test the utility of this regression model (use a two tail test with =.05). Interpret your results, including the p-value. 6. Based on your findings in 1-5, what is your opinion about using diabetes pretest to predict diabetes postest? Explain. 7. Compute the 95% confidence interval for beta-1 (the population slope). Interpret this interval. In an attempt to improve the model, we attempt to do a multiple regression model predicting diabetes posttest based on diabetes pretest and glucose. 8. Using Minitab, run the multiple regression analysis using the variables diabetes pretest and glucose to predict diabetes posttest. State the equation for this multiple regression model. 9. Perform the Global Test for Utility (F-Test). Explain your conclusion. 10. Perform the t-test on each independent variable. Explain your conclusions and clearly state how you should proceed. In particular, which independent variables should we keep and which should be discarded. 11. Is this multiple regression model better than the linear model that we generated in parts 1-7? Explain. Summarize your results from 1-11 in a report that is three pages or less in length and explains and interprets the results in ways that are understandable to someone who does not know statistics. Submission: The summary report + all of the work done in 1-11 (Minitab Output + interpretations) as an appendix. Format: A. Summary Report B. Points 1-11 addressed with appropriate output, graphs and interpretations. Be sure to number each point 1-11