ࡱ> gc ,bjbj "x[\x[\I$6, 008  a":"(((```````$Bce`('N(((`00| ` . . .(08` .(` . .Xlh[\c!l){Y*``0 aYFf*fT[[lfW\4(( .(((((``J,((( a((((f((((((((( > :  Name _________________________________  Period _______ Date ___________________ 2.1 Describing Location in a Distribution Percentiles  Definition  Distribution Examples 25th percentile90th percentile50th percentile Problem 1 More Test Scores The test scores from a particular AP Stats exam are as follows: 72 73 73 74 75 77 77 77 78 79 79 80 80 81 82 83 83 83 84 85 86 89 90 93 Construct a stemplot of the data. In what percentile does a student fall if they score an 86 on the exam? In what percentile does a student fall if they score a 72 on the exam?  Problem 2 Wins in Major League Baseball The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009.  5 9 6 2455 7 00455589 8 0345667778 9 123557 10 3 Calculate and interpret the percentiles for the Colorado Rockies who had 92 wins, the New York Yankees who had 103 wins, and the Cleveland Indians who had 65 wins. How many games did a team in the 60th percentile win? Relative Frequency and Ogives   Median Income* Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 35 to <40 1 40 to <45 10 45 to <50 14 50 to <55 12 55 to <60 5 60 to <65 6 65 to <70 3 Total 51 *Data is median household income for the 50 states and DC.   At what percentile is California, with a median household income of $57,445? What is the center of the distribution? Estimate the first quartile of the distribution. Estimate the third quartile of the distribution. What is the Interquartile Range of the distribution?  Standardized Score (z-score)  Calculation  Definition  Problem 3 - Test Scores a) Suppose the mean score on a stats exam is 80 and the standard deviation is 6.07. John scored an 86, Megan scored a 99, and Stu scored a 72. Calculate the z-score for each student. b) Suppose the mean score on a chemistry test is 76 and the standard deviation is 4. John, from part (a), scored an 82 on the chemistry exam. In which class did he do better? Explain. c) Bob scores a 79 on a calculus test where the mean score was 83. He calculates his z-score to be 1.6. How do you know he is wrong? What do you think his actual z-score d) Assuming you are correct about Bobs z-score, what was the standard deviation on the calculus exam?  Problem 4 Homerun 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, since he hit the most home runs in a single season. However, in 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. Calculate the standardized score for each player and compare. 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 In 2001, Arizona Diamondback Mark Graces home run total has a standardized score of z = 0.48. Interpret this value and calculate the number of home runs he hit.  Transforming Data  The following is a list of 2011 salaries for selected Atlanta Braves players: Player Salary (in millions) Derek Lowe 15 Chipper Jones 14 Dan Uggla 9.1 Tim Hudson 9 Michael Bourn 4.4 Martin Prado 3.1 Jason Heyward 0.497 Brooks Conrad 0.428 Freddie Freeman 0.414 Calculate the mean, the 5 number summary, IQR, and standard deviation. b) Suppose each player receives a $100,000 bonus. Show how that would change each value, and recalculate the mean and 5 number summary. What do you notice? c) Suppose each player received a 10% bonus. Show how that would change each value, and recalculate the mean and 5 number summary. What do you notice?  Adding (or subtracting) the same number to each observation  Multiplying (or dividing) each observation by the same number  Problem 5 Test Scores Here are the summary statistics for a sample of 30 test scores. The maximum possible score on the test was 50 points. n mean st dev Min Q1 Med Q3 Max Range Score 30 35.8 8.17 12 32 37 41 48 38 Suppose that the teacher was nice and added 5 points to each test score. How would this change the shape of the distribution and each of the statistics in the table above?  2.2 Density Curves and Normal Distributions Density Curves  Density Curves   Properties  INCLUDEPICTURE "http://ebooks.bfwpub.com/tps3e/figures/2_4_big.gif" \* MERGEFORMAT  Location of Mean and Median  Symmetric Skewed Right  Skewed Left  Mean as the balancing point  Uniform Distribution Example  Parameters vs Statistics Normal Distribution  Definition and Properties  Notation Problem 6 Heights of women The heights of women 18-24 years old are N(64.5, 2.5). Sketch this distribution, labeling the mean and the points one, two and three standard deviations from the mean.  Problem 7 Batting Averages In 2009, the distribution of batting averages for Major League Baseball players was approximately Normal with a mean of 0.261 with a variance of 0.01156. Write the distribution in abbreviated form. Sketch this distribution, labeling the mean and the points one, two, and three standard deviations from the mean.  The 68-95-99.7 Rule   EMBED PBrush  Problem 8 Normal vs Non-Normal The following is a data set of 72 observations. The mean is 142 and the standard deviation is 109. What percent of the observations were within one standard deviation of the mean? Two? Three?  The following is a data set of 86 SAT Writing test scores. The mean score is 583 and the standard deviation is 79. What percent of the scores were within one standard deviation of the mean? Two? Three?  Problem 9 Test Scores Suppose that a distribution of test scores is approximately Normal and the middle 95% of scores are between 72 and 84. What are the mean and standard deviation of this distribution? What percent of scores are below 75? Give an estimate  The Standard Normal Distribution  Standard Normal Distribution  z-score calculation Standard Normal Table  Problem 10 The grades on a test are normally distributed with a mean of 83 and a std dev of 5. 1) What proportion of scores were less than 70? 2) What proportion of scores were greater than 90? 3) What proportion of scores were between 70 and 90? 4) Find the proportion of scores that were between 75 and 88. 5) Find the proportion of scores that were within 1.5 standard deviations of the mean.  Normal distributions on the calculator Problem 11 Cholesterol levels for 14 year olds are N(170, 30). Use calculator to answer. What percentage of 14 year olds have cholesterol levels less than 162? greater than 240? between 170 and 240? less than 152 or more than 190?  Inverse Normal Distribution Problems  Problem 12 SAT Verbal scores are N(505, 110). a. How high must you score to be in the top 10%? Lower 10%? b. What must you score to fall in the middle 40%? Problem 13 A distribution of test scores is approximately Normal and Joe scores in the 85th percentile. How many standard deviations above the mean did he score?  Inverse Normal distributions on the calculator Problem 14 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 exceed 120 mph? b. What percent of Rafael Nadals first serves are between 100 and 110 mph? c. The fastest 20% of Nadals first serves go at least what speed? d. What is the IQR for the distribution of Nadals first serve speeds?  Normal Distribution Calculations Problem 15 According to  HYPERLINK "http://www.cdc.gov/growthcharts/" http://www.cdc.gov/growthcharts/, the heights of 3 year old females are approximately Normally distributed with a mean of 94.5 cm and a standard deviation of 4 cm. a. What proportion of 3 year old females are taller than 100 cm? b. What proportion of 3 year old females are between 90 and 95 cm? c. 80% of 3 year old females are at least how tall? d. Suppose that the mean heights for 4 year old females is 102 cm and the third quartile is 105.5 cm. What is the standard deviation, assuming the distribution of heights is approximately Normal?      PAGE   PAGE \* MERGEFORMAT 12 Key: 5|9 represents a team with 59 wins.  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