IHS AP Statistics Chapter 2 Modeling Distributions of Data MP1 Monday Tuesday Wednesday Thursday Friday August 22 A Day 23 B Day 24 A Day 25 B Day 26 A Day Ch1 Exploring Data Class Introduction Getting to Know Your Textbook Stat Survey Ch1 Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.1 Quick Check HW #1.2A: page 41 (37, 39, 43, 45, 47) HW #1.2B: page 43 (51, 53, 55, 59 62) HW Vocab Notes Ch1 Intro and 1.1 HW #1.0: page 6 (1, 3, 5, 7, 8) HW #1.1A: page 21 (11, 17, 18) HW #1.1B: page 22 (20, 22, 23, 25, 27 34) 29 B Day 30 A Day 31 B Day Sept 1 A Day 2 B Day Ch1 Exploring Data 1.2 Displaying Quant. 1.3 Describing Quantitative Data with Numbers Ch.1 Review Data with Graphs 1.1 Quick Check Quiz 1.1 and 1.2 Quiz 1.3 HW #1.2A: page 41 HW #1.3A: page 47 (69 74), page 69 (79, 81, (37, 39, 43, 45, 47) 83, 85, 86, 88, 89, 91, 93, 94a) HW: Chapter Review Exercises (p76) HW #1.2B: page 43 HW #1.3B: page 71 (97, 99, 101 105, 107 110) (51, 53, 55, 59 62) 5 HOLIDAY 6 A Day 7 B Day 8 A Day 9 B Day Ch2 Modeling Distributions of Data Labor Day FRAPPY! Practice HW: Ch1 AP Statistics Practice Test (p78) Chapter 1 Test 12 A Day 13 B Day 14 A Day 15 B Day 16 A Day Ch2 Modeling Distributions of Data 2.1 Describing Location in a Distribution HW #2.1A: page 99 (1, 5, 9, 11, 13, 15) HW #2.1B page 101 (17 23 odd, 32) 2.2 Density Curves and Normal Dists Day1 HW #2.2A: page 102 (25 30), page 128 (33-45 odd) HW #2.2B: page 129 (47 57 odd) 2.2 Density Curves and Normal Distributions Day2 HW #2.2C page 130 (54 60 even, 68 73) 19 B Day 20 A Day 21 B Day 22 A Day 23 B Day Ch2 Modeling Distributions of Data 2.2 Density Curves and Normal Distributions Day2 Ch.2 Review Chapter 2 Test HW #2.2C page 130 (54 60 even, 68 73) HW: Chapter Review Exercises (p136) HW: Ch2 AP Statistics Practice Test (p137) 26 A Day 27 B Day 28 A Day 29 B Day 30 A Day MP1 Ends Ch3 Describing Relationships 3.1 Scatterplots and Correlation 3.2 Scatterplots and Correlation - Day1 3.2 Scatterplots and Correlation Day2 1
Sec 2.1 Describing Location in a Distribution Vocabulary percentile cumulative relative frequency graphs (ogives) standardizing z-score transforming data Learning Objectives FIND and INTERPRET the percentile of an individual value within a distribution of data. ESTIMATE percentiles and individual values using a cumulative relative frequency graph. FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data. DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data. Sec 2.2 Displaying Quantitative Data with Graphs Vocabulary density curve median of a density curve mean of a density curve Normal distribution Normal curve 68-95-99.7 rule standard Normal distribution standard Normal table Normal probability plot Learning Objectives ESTIMATE the relative locations of the median and mean on a density curve. ESTIMATE areas (proportions of values) in a Normal distribution. FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence. 2
IHS AP Statistics Name: Per: Sec 2.1 Describing Location in a Distribution Identifying location in a distribution: Percentiles and z-scores Measuring Position: Percentiles (p85 86) What is a percentile? On a test, is a student s percentile the same as the percent correct? Example: Wins in Major League Baseball The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2012. 5 5 6 14 6 6899 7 234 7 569 8 113 8 56889 9 033444 9 578 Key: 6 1 represents a team with 61 wins. Problem: Find the percentiles for the following teams: (a) The Minnesota Twins, who won 66 games. (b) The Washington Nationals, who won 98 games. (c) The Texas Rangers and Baltimore Orioles, who both won 93 games. 3
Cumulative Relative Frequency Graphs (p86 88) Example: State Median Household Incomes The table and cumulative relative frequency graph below show the distribution of median household incomes for the 50 states and the District of Columbia in a recent year. Median income ($1000s) Frequency Relative frequency Cumulative frequency Cumulative relative frequency 35 to < 40 1 1/51 = 0.020 1 1/51 = 0.020 40 to < 45 10 10/51 = 0.196 11 11/51 = 0.216 45 to < 50 14 14/51 = 0.275 25 25/51 = 0.490 50 to < 55 12 12/51 = 0.236 37 37/51 = 0.725 55 to < 60 5 5/51 = 0.098 42 42/51 = 0.824 60 to < 65 6 6/51 = 0.118 48 48/51 = 0.941 65 to < 70 3 3/51 = 0.059 51 51/51 = 1.000 Problem/ (a) Interpret the point (50,.49) and (55,.725). (b) California, with a median household income of $57,445, is at what percentile? Interpret this value. (c) What is the 25 th percentile for this distribution? What is another name for this value? (d) Make a relative frequency histogram of these data. (e) Where is the original graph the steepest? What does this indicate about the distribution? Example: Who s Taller? Macy, a 3-year-old female is 100 cm tall. Brody, her 12-year-old brother is 158 cm tall. According to the Centers for Disease Control and Prevention, the heights of three-year-old females have a mean of 94.5 cm and a standard deviation of 4 cm. The mean height for 12-year-olds males is 149 cm with a standard deviation of 8 cm. Problem: Obviously, Brody is taller than Macy but who is taller, relatively speaking? That is, relative to other kids of the same ages, who is taller? 4
Measuring Position: z-scores (p89 91) How do you calculate and interpret a standardized score (z-score)? Do z-scores have units? What does the sign of a standardized score tell you? Example: Wins in Major League Baseball In 2012, the mean number of wins for teams in Major League Baseball was 81 with a standard deviation of 11.9 wins. Problem: Find and interpret the z-scores for the following teams. (a) The New York Yankees, with 95 wins. (b) The New York Mets, with 74 wins. Example: Home run kings The single-season home run record for major league baseball has been set just three times since Babe Ruth hit 60 home runs in 1927. Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in 2001. In an absolute sense, Barry Bonds had the best performance of these four players, because he hit the most home runs in a single season. However, in Year Player HR Mean SD 1927 Babe Ruth 60 7.2 9.7 1961 Roger Maris 61 18.8 13.4 1998 Mark McGwire 70 20.7 12.7 2001 Barry Bonds 73 21.4 13.2 a relative sense this may not be true. Baseball historians suggest that hitting a home run has been easier in some eras than others. This is due to many factors, including quality of batters, quality of pitchers, hardness of the baseball, dimensions of ballparks, and possible use of performance-enhancing drugs. To make a fair comparison, we should see how these performances rate relative to others hitters during the same year. Problem: (a) Compute the standardized scores for each performance using the information in the table. Which player had the most outstanding performance relative to his peers? (b) In 2001, Arizona Diamondback Mark Grace s home run total had a standardized score of z = 0.48. Interpret this value and calculate the number of home runs he hit. HW #2.1A: page 99 (1, 5, 9, 11, 13, 15) 5
2.1 Transforming Data and Density Curves Transforming Data (p92 95) What is the effect of adding or subtracting a constant from each observation? What is the effect of multiplying or dividing each observation by a constant? Example: Test scores A graph and table of summary statistics for a sample of 30 test scores are given below. The maximum possible score on the test was 50 points. Problem: Suppose that the teacher was nice and added 5 points to each test score. Below are graphs and summary statistics for the original scores and the +5 scores. How did adding +5 change the shape, center, and spread of the distribution? n x s x Min Q 1 Med Q 3 Max IQR Range Score 30 35.8 8.17 12 32 37 41 48 9 36 Score + 5 30 40.8 8.17 17 37 42 46 53 9 36 Score Score_Plus5 10 15 20 25 30 35 40 45 50 6
Example: Test scores Suppose that the teacher in the previous alternate example wanted to convert the original test scores to percents. Because the test was out of 50 points, she should multiply each score by 2 to make them out of 100. Here are graphs and summary statistics for the original scores and the doubled scores: Score Scorex2 10 20 30 40 50 60 70 80 90 100 n x s x Min Q 1 Med Q 3 Max IQR Range Score 30 35.8 8.17 12 32 37 41 48 9 36 Score 2 60 71.6 16.34 24 64 74 82 96 18 72 Problem: How did multiplying by 2 change the shape, center, and spread of the distribution? p95 97 Example: Taxi Cabs In 2010, Taxi Cabs in New York City charged an initial fee of $2.50 plus $2 per mile. In equation form, fare = 2.50 + 2(miles). At the end of a month a businessman collects all of his taxi cab receipts and analyzed the distribution of fares. The distribution was skewed to the right with a mean of $15.45 and a standard deviation of $10.20. Problem: (a) What are the mean and standard deviation of the lengths of his cab rides in miles? (b) If the businessman standardized all of the fares, what would be the shape, center, and spread of the distribution? HW #2.1B page 101 (17 23 odd, 32) 7
Sec 2.2 Density Curves and Normal Distributions Density Curves (p103 107) Example: Batting averages The histogram below shows the distribution of batting average (proportion of hits) for the 432 Major League Baseball players with at least 100 plate appearances in a recent season. The smooth curve shows the overall shape of the distribution. What is a density curve? When would we use a density curve? Why? How can you identify the mean and median of a density curve? 8
Normal Distributions (p108 109) The normal curve is often called the Gaussian distribution, after Carl Friedrich Gauss, who discovered many of its properties. Gauss, commonly viewed as one of the greatest mathematicians of all time (if not the greatest), is properly honored by Germany on their 10 Deutschmark bill. Notice the normal curve to his left. Example: How tall are 12-year old males? Problem: According to the CDC, the heights of 12-year-old males are approximately Normally distributed with a mean of 149 cm and a standard deviation of 8 cm. Sketch this distribution, labeling the mean and the points one, two, and three standard deviations from the mean. 9
Activity: For each of the approximately Normal distributions below, calculate the percentage of values within one standard deviation of the mean, within two standard deviations of the mean, and within three standard deviations of the mean. 1. Here is a dotplot showing the weights (in grams) of 36 Oreo cookies. The mean of this distribution is 11.392 g and the standard deviation is 0.081 g. 11.2 11.3 11.4 11.5 11.6 Weight (g) Mean ± 1 SD: to Mean ± 2 SD: to Mean ± 3 SD: to % within 1 SD: % within 2 SD: % within 3 SD: 2. Here is dotplot showing the scores for 50 students on an algebra test. The mean of this distribution is 76.4 and the standard deviation is 7.9. 50 60 70 80 90 100 Score Mean ± 1 SD: to Mean ± 2 SD: to Mean ± 3 SD: to % within 1 SD: % within 2 SD: % within 3 SD: 3. Here is a dotplot of Tim Lincecum s strikeout totals for each of the 32 games he pitched in during the 2009 regular season. The mean of this distribution is 8.2 with a standard deviation of 2.8. 0 2 4 6 8 10 12 14 16 Strikeouts Mean ± 1 SD: to Mean ± 2 SD: to Mean ± 3 SD: to % within 1 SD: % within 2 SD: % within 3 SD: 4. All three of the distributions above were approximately Normal in shape. Based on these examples, about what percent of the observations would you expect to find within one standard deviation of the mean in a Normal distribution? Within two standard deviations of the mean? Within three standard deviations of the mean? 10
The 68-95-99.7 Rule (p109 112) What is the 68-95-99.7 rule? When does it apply? Do you need to know about Chebyshev s inequality? Example: Batting averages In the previous alternate example about batting averages for Major League Baseball players, the mean of the 432 batting averages was 0.261 with a standard deviation of 0.034. Suppose that the distribution is exactly Normal with = 0.261 and = 0.034. Problem: (a) Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean. (b) What percent of the batting averages are above 0.329? Show your work. (c) What percent of the batting averages are between 0.193 and 0.295? Show your work. (d) How well does the 68 95 99.7 rule apply to the distribution of batting averages we encountered earlier? 11
Example: How tall are 12-year old males? Problem: (a) Using the earlier example, about what percentage of 12-year-old boys will be over 158 cm tall? (b) About what percentage of 12-year-old boys will be between 131 and 140 cm tall? Example: Test Scores Suppose that a distribution of test scores is approximately Normal and the middle 95% of scores are between 72 and 84. Problem: (a) What are the mean and standard of this distribution? (b) Can you calculate the percent of scores that are above 80? Explain. HW #2.2A: page 102 (25 30), page 128 (33-45 odd) 12
2.2 Normal Calculations The Standard Normal Distribution (p112 114) What is the standard Normal distribution? Example: Finding areas under the standard Normal curve Problem: Find the proportion of observations from the standard Normal distribution that are: (a) less than 0.54 (b) greater than 1.12 (c) greater than 3.89 (d) between 0.49 and 1.82. (e) within 1.5 standard deviations of the mean (f) A distribution of test scores is approximately Normal and Joe scores in the 85 th percentile. How many standard deviations above the mean did he score? (g) In a Normal distribution, Q 1 is how many SDs below the mean? 13
Example: Serving Speed Problem/ In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on his first serves. Assume that the distribution of his first serve speeds is Normal with a mean of 115 mph and a standard deviation of 6 mph. (a) About what proportion of his first serves would you expect to be slower than 103 mph? (b) About what proportion of his first serves would you expect to exceed 120 mph? (c) What percent of Rafael Nadal s first serves are between 100 and 110 mph? (d) The fastest 30% of Nadal s first serves go at least what speed? (e) A different player has a standard deviation of 8 mph on his first serves and 20% of his serves go less than 100 mph. If the distribution of his serve speeds is approximately Normal, what is his average first serve speed? HW #2.2B: page 129 (47 57 odd) Show work!! 14
2.2: Using the Calculator for Normal Calculations How do you do Normal calculations on the calculator? What do you need to show on the AP exam? Suppose that Clayton Kershaw of the Los Angeles Dodgers throws his fastball with a mean velocity of 94 miles per hour (mph) and a standard deviation of 2 mph and that the distribution of his fastball speeds can be modeled by a Normal distribution. (a) About what proportion of his fastballs will travel at least 100 mph? (b) About what proportion of his fastballs will travel less than 90 mph? (c) About what proportion of his fastballs will travel between 93 and 95 mph? (d) What is the 30 th percentile of Kershaw s distribution of fastball velocities? (e) What fastball velocities would be considered low outliers for Kershaw? (f) Suppose that a different pitcher s fastballs have a mean velocity of 92 mph and 40% of his fastballs go less than 90 mph. What is his standard deviation of his fastball velocities, assuming his distribution of velocities can be modeled by a Normal distribution? HW #2.2C page 130 (54 60 even, 68 73) 15
2.2 Assessing Normality (p121 122) Example: No space in the fridge? The measurements listed below describe the useable capacity (in cubic feet) of a sample of 36 side-by-side refrigerators. (Source: Consumer Reports, May 2010). Problem: Are the data close to Normal? 12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.3 15.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.4 16.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4 p122 125 When looking at a Normal probability plot, how can we determine if a distribution is approximately Normal? Sketch a Normal probability plot for a distribution that is strongly skewed to the left. Example: State areas Problem: Use the histogram and Normal probability plot below to determine if the distribution of areas for the 50 states is approximately Normal. 16