MATH 131 - Unit 2 Class Activity 2 8/16 Expected value is an important concept to consider when calculating premiums for insurance. The insurance company will assign probabilities to the possible outcomes for the group based on research data. The company will then use expected value to determine the premium to charge. Example 1: Dianne is a 25 year old teacher in good health. Reviewing actuarial tables and based on her age, gender and health, her insurance company has proposed that the probability that she will die this year is 0.001. (a) What is the probability that she will survive the year? (b) If she buys a $100,000 life insurance policy, what are the two possible values the company may pay? (List then in the x column below.) (c) Put the correct P(x) value next to your outcomes. (d) Find the expected value for probability distribution. (e) If the company charges a $100 service fee, what should be Dianne's annual premium? x P( x) x P( x) 1 MATH 131 - Unit 2 Class Activity 2 8/16 Example 2: Dan is pricing car insurance from a new company that only pays out at 3 levels: total, major accident and fender bender. His car is valued at $20,000. After reviewing his application, his insurance company assigned him to a group with the probabilities listed below. These values are used to figure the value the company must charge to cover their payouts. If the company charges a $200 service fee, what will be his annual premium? x , payout P( x) 20,000 total 0.010 10,000 major 0.020 2,000 fender bender 0.090 0 no accident x P( x) ? Try it yourself: If you were buying a similar policy with your car also valued at $20,000, what probabilities for payout do you believe would be assigned. What should be your annual premium? x , payout P( x) 20,000 total 10,000 major 2,000 fender bender 0 no accident 2 x P( x) MATH 131 Lab 3 8/16 The goal of this lab is to find descriptive statistics for your quantitative data. Part 1: V2, Quantitative Data (3 points) Use the list of all 40 numbers from Lab 1, column V2 for this part of the lab. A. Identify the name of the quantitative variable V2 that you are using _Peak Position. B. Use StatCrunch or Minitab to find Mean Median Mode Standard Deviation Minimum Value Maximum Value Q1 Q3 C. Using the appropriate numbers from B above, make a box-and-whisker plot for your data. What does this plot tell you about your data? (2 Points) D. Looking at your list in B, list your Minimum, Maximum and Mean data values. List below and then find the z-score for each. Data Value, x z-score ( z= xx ) s Minimum Maximum Mean E. How do -scores indicate a data value is above the mean? _______________________. z below the mean? _________________________. What -score always corresponds to the mean? ___________________________. z 1 MATH 131 Lab 3 8/16 (2 Points) F. Use Chebychev's Theorem to write a statement about the interval from 2 standard deviations below the mean to 2 standard deviations above the mean. List the specific interval you used for your data. Interpret the result in the context of your data. Part 2: V3, Quantitative Data (3 Points) A. What is the name of your V3 variable? ________________________________ B. Fill in the values in the chart below using the frequency distribution in Lab 2, Part 3. You may not need all of the rows below. C. Use the appropriate values in the table to find the mean for your grouped data. Show your computations. Remember Data Groups, using V3 (x f ) x = Frequenc y f Class Midpoint n x f =n D. Use StatCrunch or Minitab to find the mean and standard deviation of the raw data. (from Lab 1). E. Compare the answers in C and D. What accounts for any difference? StatCrunch directions Minitab directions 1. Stat 2. Summary stats 3. Columns 1. Stat 2. Basic Stat 3. Display descriptive statistics 2 xf (xf ) MATH 131 Lab 3 8/16 3 Billboard Top 100 Artists Artist/Group Migos Ed Sheeran Peak Position Weeks on Chart Movement Direction Random Number Information (from previous week) 1 17 up 3 Billboard Music Chart (Top 100 Artists) 1 130 same 4 http://www.billboard.com/charts/artist 100 Brantley Gilbert 3 42 up The Weeknd 1 121 down Drake 1 136 down 15 up means the artist moved up in the rankings from the previous week 5 For week of February 18, 2017 Bruno Mars 1 124 down The Chainsmokers 2 62 down 16 down means the artist moved down in the rankings from the previous week 17 same means the artist didn't move in the rankings from the previous week Rihanna 2 132 down Taylor Swift 1 132 up 23 twenty one pilots 1 94 down 25 Kehlani 11 6 none 28 Adele 1 105 down 29 Shawn Mendes 1 104 down 32 Train 14 7 up 33 Ariana Grande 1 134 down 34 Zayn 1 41 up 39 Alessia Cara 12 74 down 40 Sam Hunt 5 129 up 41 Maroon 5 1 136 down 43 New Edition 20 2 up 45 Big Sean 2 78 down 46 Justin Bieber 1 135 down 48 J. Cole 1 95 down 50 Rae Sremmurd 5 100 down 55 Beyonce 2 134 up 57 Lil Uzi Vert 16 31 down 61 Sia 5 136 down 64 Bell Biv Devoe 28 1 none 67 Metallica 2 86 down 74 Jon Bellion 25 17 down 75 Niall Horan 11 19 up 77 Camila Cabello 29 19 up 78 Marian Hill 33 3 up 79 Eminem 11 136 down 86 John Legend 15 90 down 88 Amine 27 16 up 91 Justin Timberlake 5 90 same 92 Keith Urban 8 97 down Panic! At The Disco 3 63 down 99 Gucci Mane 6 17 up 100 9 21 none means the artist was not ranked in the Top 100 the previous week 96 1 MATH 131 Lab 2 8/16 The goal of this lab is to organize your data from Lab 1 into frequency distributions and graphs. Graphs should be made by hand, using a ruler. Part 1: V1, Qualitative Data (3 points) A. Prepare a frequency and relative frequency distribution for your qualitative data V1 in the format noted below. Fill in the appropriate data values and frequencies for your data from Lab 1. V1:_Movement Direction Qualitative categories Up Frequency, f 13 Relative frequency, rf .325 Down Same None 23 2 2 .575 .05 .05 B. Prepare a Pareto chart for your data. 25 20 15 23 10 13 5 0 Down Up Series 3 1 2 2 None Same MATH 131 8/16 Lab 2 C. Write a one-sentence explanation about what these charts tell you about your data and whether you expected this result. The data shows that most of the songs went down, same or no movement in direction was pretty close. Also, movement direction going up was a little less than movement direction going down. Part 2: V2, Quantitative Data (3 points) A. List the name for V2, the quantitative variable for this part of the lab. Organize your data into a frequency distribution using between 5 and 10 groups or single values, as appropriate for your data. V2:_Peak Position_________________ V2 values or V2 data groups 1-10 11-20 21-30 31-40 If data groups are used, list class midpoints Frequency, f 27 8 4 1 Relative frequency, rf .675 .20 .125 .025 B. Prepare a relative frequency histogram for your data. Be sure to label clearly. 2 MATH 131 Lab 2 8/16 Chart Title 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 thru 10 11 thru 20 21 thru 30 30 thru 40 Column3 C. Comment about your histogram. Use words like "symmetric", "skewed left or right", as appropriate. It's skewed to the right of the graph because it goes down as the numbers get higher. Part 3: V3, Quantitative Data (4 points) A. Prepare a frequency distribution for your data, using between 5 and 10 groups. V3:___Weeks on the Chart_______________ V3 data groups 1-20 21-40 41-50 51-70 71-90 91-110 111-130 131-150 Class midpoints 10.5 30.5 40.5 60.5 80.5 100.5 120.5 140.5 3 Frequency, f 6 1 3 3 5 8 4 10 MATH 131 Lab 2 8/16 B. Prepare a frequency polygon using class midpoints. Label carefully. Weeks on the chart 40 35 30 25 20 15 10 5 0 10.5 30.5 40.5 60.5 80.5 100.5 120.5 Column3 C. What does the graphs communicate to you about your data? 10 songs have been on the charts for the highest amount of weeks. 4 140.5 MATH 131 - Unit 2 Class Activity 2 8/16 Expected value is an important concept to consider when calculating premiums for insurance. The insurance company will assign probabilities to the possible outcomes for the group based on research data. The company will then use expected value to determine the premium to charge. Example 1: Dianne is a 25 year old teacher in good health. Reviewing actuarial tables and based on her age, gender and health, her insurance company has proposed that the probability that she will die this year is 0.001. (a) What is the probability that she will survive the year? Pr(she will survive) = 1-0.001 = 0.999 (b) If she buys a $100,000 life insurance policy, what are the two possible values the company may pay? (List then in the x column below.) The company will pay; $100,000 with probability 0.001, or $0 with probability 0.999. (c) Put the correct P(x) value next to your outcomes. (d) Find the expected value for probability distribution. The expected value of the insurance payment is p(x) = $100,000*0.001 + $0*0.999 = $100 (e) If the company charges a $100 service fee, what should be Dianne's annual premium? The insurance company will charge a premium = the sum of the service fee and their expected payment value = $100 + $100 = $200 . . . Diane's annual premium x P( x) x P( x) 1 MATH 131 - Unit 2 Class Activity 2 8/16 $100,000 0.001 $100 $0 0.999 $0 $100 Example 2: Dan is pricing car insurance from a new company that only pays out at 3 levels: total, major accident and fender bender. His car is valued at $20,000. After reviewing his application, his insurance company assigned him to a group with the probabilities listed below. These values are used to figure the value the company must charge to cover their payouts. If the company charges a $200 service fee, what will be his annual premium? x , payout P( x) x P( x) 20,000 total 0.010 $200 10,000 major 0.020 $200 2,000 fender bender 0.090 $180 no accident 0.88 $0 0 The annual premium = $200+$200+ $180+$200 = $ 780 Try it yourself: If you were buying a similar policy with your car also valued at $20,000, what probabilities for payout do you believe would be assigned. What should be your annual premium? x , payout P( x) x P( x) 20,000 total 200/20000 = 0.01 $200 10,000 major 200/10000 = 0.02 $200 2,000 fender bender 200/2000 = 0.1 $200 0 no accident 0.13 2 0 MATH 131 - Unit 2 Class Activity 2 8/16 The premium = $200+$200+$200 +$200 = $800 3