Tutor Marked Exercise 3 I. Theory Component Where relevant, keep all your middle work as well as your final answer to a minimum of four decimals, unless otherwise stated. 1. Carefree Hospital is a government funded institution that operates in a small Canadian city. Carefree recorded the number of daily emergency visits (rounded to the nearest 50) for each of the past 100 days as shown below, in order to better plan its emergency service manpower needs, e.g., number of emergency room nurses, admin staff, and doctors to hire and schedule. X = Number of daily emergency visits Frequency in terms of number of days each value of X has occurred. 200 40 250 30 300 20 350 10 a. Let X be the number of daily emergency visits. Construct a probability distribution for X, based on an analysis of the past 100 days. [4 marks] b. Based on the probability distribution, constructed in part a. above, compute the mean number of emergency visits per day. [3 marks] c. Based on the probability distribution, constructed in part a. above, compute the variance and standard deviation of X. [6 marks] d. One of the manpower goals of the hospital is to be as efficient as possible in staffing its emergency rooms. An efficient situation would be where the hospital hires just enough staff so that each staff member is being fully utilized (working to capacity). If the hospital plans its staffing based on the mean # visits calculated in part b. above, under which case below is the goal of manpower efficiency more difficult to achieve. Explain. [3 marks] Case 1: The standard deviation is relatively high. Case 2: The standard deviation is relatively low. 2. Butt-Out is a new "stop smoking" prescription drug that was recently approved by the government. Past clinical trials indicate that 70% of regular smokers who take this new drug quit smoking within a month. A local doctor has just prescribed Butt-Out to ten of her patients who are regular smokers. Find the probability that a. all ten patients will quit smoking within a month. [4 marks] b. at most 6 of the patients will quit smoking within a month. [4 marks] c. at least 1 of the patients will quit smoking within a month. [4 marks] d. more than half of the ten patients do NOT quit smoking within a month. [5 marks] 3. According to its product specifications, after being fully charged, the Mobile X Laptop computer's battery operating life in hours is normally distributed with a mean operating life (before needing recharging) of 120 minutes and a standard deviation of 50 minutes. As a Mobile X laptop user, you are about to use this fully charged laptop to complete an exam. Find the probability that your Mobile X Laptop computer will have a battery operating life of: a. between 80 and 150 minutes. [5 marks] b. at most 100 minutes. [5 marks] c. at least 150 minutes (which is the duration of the exam). [5 marks] 4. A nursing professor is currently analyzing the scores of a final exam she administered to her students. The scores are normally distributed with a mean of 25 and a standard deviation of 5. Between what two scores does the middle 50% of her students lie? [6 marks] 5. Refer to the previous question to answer this question. Because of the low final exam scores, the professor has decided to scale (often called "curve") the scores based on the normal distribution, as follows. She wants to award: a. an "A" grade to those scores within the top 10% of the scores. b. a "B" grade to those scores above the bottom 70% and below the top 10% of the scores. c. a "C" grade to those scores above the bottom 30% and below the top 30% of the scores. d. a "D" grade to those scores above the bottom 10% and below the top 70% of the scores. e. an "F" failing grade to those scores within the bottom 10% of the scores. i. Find the numerical limits of the exam scores that qualify for an "A" grade. [5 marks] ii. Find the numerical limits of the exam scores that qualify for a "B" grade. [5 marks] iii. Find the numerical limits of the exam scores that qualify for an "F" grade. [5 marks] 6. An insurance company charges its policy holders an annual premium of $200 for the following type of injury insurance policy. If the policy holder suffers a "major injury" resulting in lengthy hospitalization, the company will pay out $15,000 to the injured policy holder. If the policy holder suffers a "minor injury" resulting in significant absence from work, the company will pay out $4,000 to the injured policy holder. If no injury is encountered (the most probable event) the company, of course, does not payout anything to the policy holder. Past records show that each year, 1 in every 2000 policy holders experience a "major injury" and 1 in every 500 experience a "minor injury.." Assuming that the only company expense related to this policy is the annual payout. a. Construct a probability distribution table for "X" where "X" refers to the annual profit for this policy, where "X" = Annual Premium - Annual Payout. [6 marks] b. Compute the expected annual profit that the company can expect to receive per policy holder. [5 marks] II. Computer Component Where relevant, do NOT round off the results you get from Minitab. Make sure that for each computer problem: you paste your results, as requested, to the Minitab Report Pad window AND that you save to the ONE Minitab project fileTME_Unit3. 7. Consider the following simplified version of the numbers game operated many years ago by the Mafia. Once you pay $1.00 you pick any number between 0 and 99 inclusive. If the number you picked is randomly selected by a gambling hall operator, called the dealer, you win the game. Before the numbers game begins you are given odds of 49-1. This means that if you win the game once, you make a profit of $49.00, if you bet (pay) $1.00. In other words, the dealer will pay you back $50.00, each time you win. Since it costs you $1.00, this amounts to a $49.00 profit per game won. If you lose the game, you lose your dollar (profit is $-1.00) We will define the random variable, X, to be the profit amounts related to the different events possible each time you play the numbers game. Based on this definition, we can form the following table, which describes each value of the random variable, the probability distribution, and the expected value associated with the numbers game. Event For Each Play X = Profit P(X) XP(X) Win $49 1/100 = 0.01 49 x .01= $.49 Lose $-1 99/100 = 0.99 -1 x .99 = $-.99 1.00 E(X) = $-.050 Total a. Download the Minitab Project file called TME_Unit3 from the appropriate Assignment Drop Box and save it to your hard drive or memory device. Open the TME_Unit3 project and create a Minitab worksheet (within this project file) called Numbers, which you will use for the following problem. Use Minitab to simulate your 1000 plays of the numbers game by generating 1000 integer random numbers in Column C1 of the worksheet named Numbers. Each number generated will be between 0 and 99. After generating Column C1, copy and paste the first 5 rows of Column C1, from the Numbers Worksheet to Report Pad in the Minitab's Project Manager window in the space just under question 7a. Save your results in the Minitab Project file called SelfTest_Unit3. [1 mark] b. Suppose that you always bet the lucky number "7" for each of the 1000 numbers plays. This means that any time the random number generated in Column C1 is a "7" you achieve a profit of $49. Anytime the random number is NOT a "7" you achieve a profit of $-1. Use Minitab to recode the original random numbers in Column C1 as new numeric profit values stored in Column C2 as follows: After generating Column C2, copy and paste the first 5 rows of Column C1 and C2 from the Numbers Worksheet to Report Pad in the Minitab's Project Manager window in the space just under question 7b. Save your results in the Minitab Project file called TME_Unit3. [2 marks] Original Values (random numbers) New Values ("profit") 0:6 (0 to 6) -1 7 49 8:99 (8 to 99) -1 c. Use Minitab to estimate your expected profit from playing the numbers game 1000 times by computing the arithmetic mean of the 1000 profit values in column C2. Display your result in the Minitab session window. Note that each time you generate another 1000 plays of this game, you will get a different expected profit. Copy and paste the mean profit you recorded from the Session Window to the Report Pad Window under question 7c. Save your results in the Minitab Project file called TME_Unit3. [1 mark] 8. A surgical technique is about to be performed on 60 patients. Past records show that the chances of success are 70%. Use Minitab to compute the binomial probability that the surgery being performed on the 60 people will be successful for: a. all 60 patients Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 8a. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [1 mark] b. at most 30 people Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 8b. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] c. at least 45 people Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 8c. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] d. less than 40 people Type in the specific answer the appropriate blank space provided in theReport Pad Window under question 8d. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] 9. Suppose that the time per workout that a professional athlete works on stretching exercises is normally distributed with a mean of 10 minutes and a standard deviation of 4 minutes. Find the probability that on the next workout a randomly selected professional athlete will spend a. at most 8 minutes on stretching exercises. Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 9a. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] b. more than 15 minutes on stretching exercises. Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 9b. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] c. between 9 and 12 minutes on stretching exercises. Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 9c. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [2 marks] 10.Refer to problem 9 above. Eighty percent of professional athletes who work out will spend more than what amount of time on stretching exercises? Use Minitab to answer this question, which is asking you to find a normal value given a probability (or percentage). Type in the specific answer the appropriate blank space provided in the Report Pad Window under question 10. Below this answer, copy and paste the related results from the Minitab Session Window to the Report Pad Window. Re-save the Minitab Project file TME_Unit3 so that all your answers above are saved in this project file. [3 marks]