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Honors Stats Day 13 Quiz and Normal Dist SILENT DO NOW ON DESK: Chapter 5 Practice Quiz DO NOW: Check HW with key and grading pen Homework: Ch 6 #17-19 Quiz Chapter 5 • Making Boxplots • Reading Boxplots (SOCS and Interpret) • Mean vs Median • Standard Deviation Boxplot Exit Ticket NEVER INCLUDE Q2 • (Based on AP Test) • QUIZ + Calculator Sheet • • Pass in Quiz and Practice Calculator Work with the following data set: 5, 6, 12, 12, 12, 15, 18, 20, 22, 56 Random Data • • • Remember: Data that is completely random (no bias) should present with a NORMAL DISTRIBUTION among the population Ex. Heights of men, the “bell curve” A Normal Distribution Standard Deviation MEAN Drawing Normal Models • Notation: N ( mean, standard dev ) • Example: N(18,3) Sketch the Normal Distribution 1) N(500, 100) • 2) N(100, 16) • Honors Stats Day 14 Review SILENT DO NOW ON DESK: Chapter 6 #17a-19a DO NOW: ACT Do Now Check HW with key and grading pen Homework: Chapter 6 #17-19, Chapter 6 Notes A Normal Distribution Standard Deviation MEAN Drawing Normal Models • Notation: N ( mean, standard dev ) • Example: N(18,3) Sketch the Normal Distribution 1) N(500, 100) • 2) N(100, 16) • Knowing Extremes Normal models show how LIKELY it is to find a value that far from the mean or what percentage of the data fell below, above, or between given value(s) Sketch and Describe • • • • Sketch the distribution of the birthweights of babies: N(7.6lb, 1.3lb) We are 68% sure that the baby will be born between ______lb and ______lb. We are 99.7% sure that the baby will be born between _____lb and _____lb. What is the likelihood that the baby will be less than 8.9 lbs? You try • • The distribution of body temperatures is modeled normally as N(98.2, 0.6). Describe the distribution. What is the probability that your body temp is below 97 degrees? Unusual • We say something is “unusual” if it is more than 2 standard deviations from the center/mean (from the “norm”) Worksheet and Board Practice Honors Stats Day 15 SILENT DO NOW ON DESK: Chapter 6 #17-19, notes DO NOW: Check HW with key and grading pen 68-95-99.7 Exit Slip Homework: Ch 6 #5-7, 9, 29, 30, 33, 34 Objective • SWBAT calculate probability using the standard normal curve and zscore chart Last Week… We have 2 problems with this.. • 1) How do we use this model to find probability that is not exactly ±1, ±2, or ±3, SDs away from the mean? • • 2) Can we use this to compare two completely different data sets?? Notice… • We have been talking in terms of “Number of Standard Deviations Away from the Mean”… Another way to see it… • • Let’s look at it as number of standard deviations away from the center (the mean)… And define it as the Z-SCORE! The Normal Distribution Notice how each line represents the number of standard deviations away from the mean..or the… Z-SCORE!!! Standard Deviation MEAN 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?? We shift everything over (mean becomes 0) We subtract EVERY VALUE by that number What does not change?? SPREAD: Standard Deviation, Range, IQR 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 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! Comparisons • Compare earning a 21 on the ACT N(20.8, 2.4) to earning a 1200 on the SAT N(1020, 184). Which one did the person do better on? Comparisons Who’s more respected in their workplace based on pay: • A teacher who is paid $70,000 when the avg teacher pay at the school is $45,000 with an SD of $9,000 or a consultant who earns $92,000 when the average is $80,000 with an SD of $5,000. • Comparisons • Chapter 4 Quiz P2: Mean = 19.2 SD = 0.5 Maria got a 19 Chapter 4 Quiz P3: Mean = 19.2 SD = 1.8 Leslie got an 19 Who did better on the quiz compared to their class? Comparing 2 Different Data Sets • We need a way to compare 2 different data sets taking into account BOTH the… average • standard deviation • If we make their averages the same (0), we can use the SD as a way of measuring… hence the Z-SCORE One More: Which is fatter? • • • Which is fatter, the cat or the dog? The cat weighs 13lbs and cats on average weigh 9lbs with a standard deviation of 2.1 lbs. The dog weighs 72 lbs when they on average weigh 50 lbs with a standard deviation of 15.8lbs. If it’s not exactly ±1, ±2, or ±3 SDs from the mean • • Consdier the SAT again: N(1020, 184) Suppose you earned a 1200 on the SAT. What percentile are you? (PERCENTILE=%THAT YOU BEAT!) 1. Find Z score 2. What percent is to the LEFT?? THE CHART ONLY GIVES YOU WHAT IS ON THE LEFT! • • N(1020,184) Suppose you earned a 1200 on 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! Z-SCORE CHART 1. 2. 3. 4. Find your z-score Draw a picture- vertical line at zscore 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) Z-Score • 1. What is the likelihood that you scored below an 18 on this last ACT given N(18.8, 3.7)? • 2. That you scored above a 20? • 3. That you scored above a 22.5? • 4. What percentile are you if you scored a 24? Honors Stats Day 16 SILENT DO NOW ON DESK: Chapter 6 #5-7, 29, 30, 33, 34 DO NOW: Check HW with key and grading pen Homework: Quick Review 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? • 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! Finding the Between • Find the percentages for each and subtract the larger from the smaller! Practice again! • What is the chance that a baby weighs between 4.5 oz and 8.5 oz given N(6, 1.7)? Board Practice 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 Question 1 • What percentile am I if I scored a 25 on the ACT that is normally distributed with N(21, 3.2)? 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)? 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) Question 4 What percent of students score between a 21 and a 26 on the ACT that is normally distributed • N(21, 3.2)? • Question 5 • What percent of M&M packages weigh between 2.4 and 2.6 ozs given N(2.43, 0.4)? Question 6 • What percent chance is there that you will score more than 2 goals in a game given N(2.4, 0.8) 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? 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? 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? Question 10 • Who is a better choice for the ad campaign if Ronaldo attracts an average of 1 million viewers with an SD of .023 million while Lebron James attracts an average of .9 million with an SD of .011 million? Question 11 Which is a better deal: a $16000 yamaha motorcycle which averages • $15000 with an SD of $1900, or a$23000 Harley motorcycle which averages $21000 with an SD of $2400? • CHALLENGE • I got in the 95th percentile for the SAT N(1020,184). What was my score on the SAT? 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 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? 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? Exit Ticket