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Stats Day 22

Learn how to calculate likelihood using a z-score chart and determine missing values given percentiles and the normal distribution. Practice with examples and explore the standard normal model.

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Stats Day 22

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  1. Stats Day 22 More on Z-Scores

  2. SILENT DO NOW DO NOW: ACT Half Sheet HW due TOMORROW: Ch. 6 #17-19, 29, 33, 34

  3. Objective • SWBAT calculate likelihood/probability using a z-score chart • SWBAT calculate a missing value given its likelihood/percentile and the normal distribution (mean, sd)

  4. Practice 68-95-99.7 • On the Boards!

  5. Example 2(not on 68-95-99.7) • What if it is not EXACTLY 0, ±1, ±2, or ±3 standard deviations away from mean, how do we calculate probability? • This is where we use the • STANDARD NORMAL MODEL

  6. The Normal Distribution Notice how each line represents the number of standard deviations away from the mean..or the… Z-SCORE!!! MEAN Standard Deviation

  7. The Standard Normal Curve • The Distribution of Z-SCORES! THERE ARE NO UNITS! It’s the same for every set of data, no matter what you are talking about!

  8. Z-score as a shift and rescale of distribution • When we make a z score, we take each data point and subtract a number (the mean) • What would happen to the graph y=x if we subtracted 1 from each x value y=(x-1)? • What changes?? • What does not change?? We shift everything over (mean becomes 0) We subtract EVERY VALUE by that number SPREAD: Standard Deviation, Range, IQR

  9. Shift and Rescale • When we make a z score, we also take each data point and divide by a number (the standard deviation) CHANGING OUR SCALE TO STANDARD DEVIATION

  10. The Standard Normal Curve • The Distribution of Z-SCORES! THERE ARE NO UNITS! It’s the same for every set of data, no matter what you are talking about! YAY!

  11. How can this help us? • N(500,100) • Suppose you earned a 600 on one part of the SAT. What percentile are you? 1. Find Z score 2. What percent is to the LEFT?? We also have a chart! Since there are no units, we can use it for everything!

  12. Yesterday… • Focused on Normal Distribution, but saw that we could shift the normal distribution to show a distribution of z-scores: the Standard Normal Curve • Today… • We will use z-scores the Standard Normal Curve Chart to determine percent and probability

  13. If it’s not exactly ±1, ±2, or ±3 SDs from the mean • If your z-score is not exactly ±1, ±2, or ±3, • The chart can tell you the percent! • N(500,100) • Suppose you earned a 680 on one part of the SAT. What percentile are you? 1. Find Z score 2. What percent is to the LEFT?? THE CHART ONLY GIVES YOU WHAT IS ON THE LEFT!

  14. N(500,100) • Suppose you earned a 720 on one part of the SAT. What percent of test takers did better than you? 1. Find Z score 2. What percent is to the RIGHT?? (Draw a picture) The percent the chart will give you is the percent to the left… To get the right, subtract from 100!

  15. Z-SCORE CHART • Find your z-score • Draw a picture- vertical line at z-score and shade • Look up z-score to tell you percent to the LEFT of your score (less than your score) Note: this is the percentile • Determine if you want that percent (less than) or 100-[that percent] (more than)

  16. Z-Score • 1. What is the likelihood that you scored below an 18on this last ACT given N(18.8, 3.7)? • 2. That you scored above a 20? • 3. That you scored above a 22.5?

  17. Find the following in pairs • 1) z < 2.88 • 2) z<-0.11 • 3) z > 0.24 • 4) -1.31 < z • 5) Drink prices at a coffee shop are distributed normally N(3.25, 1.00). What is the probability that your drink will cost more than $5?

  18. Challenge • Life span is distributed normally N(71, 14). What is the probability that you will live to at least 75 but no older than 90? • Hint: Draw a Picture! • Find the percentages for each and subtract the larger from the smaller!

  19. Finding the Between • Find the percentages for each and subtract the larger from the smaller!

  20. Practice again! • What is the chance that a baby weighs between 4.5 oz and 8.5 oz given N(6, 1.7)?

  21. Stats Day 23 Final Z-Score Game

  22. SILENT DO NOW ON DESK: Chapter 6 #17-19, 29, 33, 34 DO NOW: Check HW with Grading Pen Homework: Bring Book on Monday Due TUESDAY Ch. 6 #31, 32, 36, 38, 41, 43 **Ch 6 Quiz Tuesday

  23. Quick Review • **DRAW PICTURE • P3 Class grades are normally distributed with a mean of 83 and a standard deviation of 5.1. • What is the probability that a new kid entering my class will… • 1) get below an 80%? • 2) Above a 92%? • 3) Between an 85 and a 95%?

  24. Trashketball! • You can use your notes • Everyone in your group must have an answer • To get the shots, everyone in group raises hands with completed work raised hand and not done? Disqualified from round • After each problem, the winning group will show work

  25. Question 1 • What percentile am I if I scored a 25 on the ACT that is normally distributed with N(21, 3.2)?

  26. Question 2 • What is the chance that I scored above a 93 on a test that was normally distributed as follows: N(88, 3.6)?

  27. Question 3 • What percent of basketball players have less than a .6 shooting percentage at the free-throw line if it is normally distributed N(.71, .09)

  28. Question 4 • What percent of students score between a 21 and a 26 on the ACT that is normally distributed • N(21, 3.2)?

  29. Question 5 • What percent of M&M packages weigh between 2.4 and 2.6 ozs given N(2.43, 0.4)?

  30. Question 6 • What percent chance is there that you will score more than 2 goals in a game given N(2.4, 0.8)

  31. Question 7 • Chip bag weights are normally distributed with a mean of 1.8 oz with a SD of .1oz. What is the probability of getting a chip bag that weighs 2 oz or more?

  32. Question 8 • Labrador retriever life spans are normally distributed N(11.8, 1.3). What is the probability that your chocolate lab lives between 14 and 15 years?

  33. Question 9 • The average room temperature in Pritzker is 70 degrees Fahrenheit with an SD of 5 degrees. What is the probability that your next room will be between 68 and 72 degrees?

  34. Exit Ticket

  35. Finding scores from percents • Say a school only accepts students in the top 10%(90th percentile). If the ACT is normally distributed N(21, 3.2), what score would you need to get on your ACT? WORK BACKWARDS 1. Draw a picture 2. Determine percent to the left 3. Find z in chart from percent 4. Plug in equation and solve for x

  36. Another Example • The organic chemistry class at Boston College, after their first test, suggests to its lowest 15% of students that they should drop the course. If the class has a distribution of N(78, 8.2), what score would you have to be above to stay in the class?

  37. Another Example • My friend just had a premature baby that was born in the 30th percentile of baby weights. If newborns have a normally distributed weight modeled by N(7.5lbs, 1.25lbs), how much did my friend’s baby weigh?

  38. Stats Day 24 Ch 6 Review

  39. SILENT DO NOW DO NOW: ACT Halfsheet Homework: Due TOMORROW Ch. 6 #31, 32, 36, 38, 41, 43 **Ch 6 Quiz TOMORROW

  40. Chapter 6 Topics • Definition and formula of Z-Score • Z-Score to compare two different data sets • Normal Curve and 68-95-99.7 rule • Standard Normal Curve • Definition of percentile • Z-score to find probability/percentile • Less than • Greater than • Between • Given probability/percentile, find x • Any relevant notation

  41. Sample Study Guide

  42. In Class Study Guide and Practice Quiz

  43. Stats Day 25 Ch 6 QUIZ

  44. Finding z-score on calculator!

  45. Unit 1 Group Teach • Each group will re-teach objectives to the class • You must teach objectives with key points and at least 2 examples • You will have max 10 min to teach • Begin prep with your group NOW!

  46. Homework • Unit 1 Review Packet DUE in 3 DAYS • This will be “Schrishuhn Style” • You must complete the entire packet. • I will assign a problem to students randomly and all problems will be put on the board. • When you finish, check with me if it is correct. If you are correct you may do a room walk/visit others to check the rest of your answers and ask questions. (I suggest doing corrections in another color). Remember, this is studying for your test! • When you have checked and visited you may sit down and try some extra credit!

  47. Stats Day 22 Group Teach

  48. SILENT DO NOW ON DESK: Group Teach Notes from Yesterday DO NOW: Spend 10 minutes preparing with partner for Group Teach

  49. Group Teach

  50. Homework • Unit 1 Review Packet DUE in 2 DAYS • This will be “Schrishuhn Style” • You must complete the entire packet. • I will assign a problem to students randomly and all problems will be put on the board. • When you finish, check with me if it is correct. If you are correct you may do a room walk/visit others to check the rest of your answers and ask questions. (I suggest doing corrections in another color). Remember, this is studying for your test! • When you have checked and visited you may sit down and try some extra credit!

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