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Name _______________________________ AP STATISTICS CHAPTER 2: Consider the following data and graph concerning heights of freshmen boys: Heights of Freshman Males HeightIn... Percent... <new > = 1 64 3 2 65 8 3 66 13 4 67 14 5 68 16 6 69 15 7 70 11 8 71 8 9 72 4 10 73 4 11 74 2 12 75 1 4 13 76 1 2 Line Scatter Plot Heights of Freshman Males 18 16 PercentOfFreshmen 14 12 10 8 6 0 64 What percent of freshmen have heights below 70 inches? What is the median height of a freshman boy? DEFINITION: The pth percentile of a distribution is the value such that p percent of the observations fall at or below it. What is the 30th percentile of this distribution? What is the 90th percentile? At what percentile would a student of height 71 inches fall? An OGIVE, or a relative frequency graph allows us to view the relative standing of individual observations. The y-axis in this type of graph tells the relative cumulative frequency for the distribution. Create an ogive for the height of freshmen males data set. SUMMARY/QUESTIONS TO ASK IN CLASS 66 68 70 72 HeightInches 74 76 78 Name _______________________________ AP STATISTICS CHAPTER 2: Class Notes: You will download a list of salaries of CEO’s from 59 small businesses (in thousands of dollars) into your calculators. Give the mean and standard deviation of this data, and create a boxplot. Describe the shape of the distribution. Consider the following changes to the data. Use your graphing calculator to again determine the mean and standard deviation, and describe the shape of the distribution. Create side-by-side boxplots to display the distribution. 1. The salary of each CEO is increased by 80 thousand dollars. (Use L2) 2. The salary of each CEO rises 30%. (Use L3) 3. The salary of each CEO falls 40%, but then rises $20,000. (Use L4) How do changes to a data set, through addition, subtraction, or multiplication, affect the mean and standard deviation of a distribution? LINEAR TRANSFORMATION: A linear transformation changes the original variable x into the new variable xnew given an equation by the form: xnew = a + bx . Effects of a: Effects of b: Effects on the shape of the distribution: SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: DENSITY CURVES AND THE NORMAL DISTRIBUTION The histogram to the right represents the final grades for Mr. E’s Calculus classes. 1. What proportion of students had scores below 70? 2. What proportion of students had scores between 75 and 100? As data sets become larger, we can draw a curve which describes the overall pattern of the distribution. 1. About what proportion of this curve lies below 70? 2. About what proportion of this curve lies between 70 and 75? 3. If the proportions represented by each bar in the histogram were equal to the area of each bar, then what is the total area under this curve? Steps in exploring quantitative data: Density Curves: SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: SYMMETRIC AND SKEWED DISTRIBUTIONS Locate the mean and median in these density curves: NOTATION FOR MEAN AND STANDARD DEVIATION Mean Standard Deviation SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: NORMAL DISTRIBUTIONS Words which describe a normal distribution: The density curve for a normal distribution is described by giving The Empirical Rule: also known as The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55). a) Sketch the Normal density curve for this distribution. b) What percent of ITBS vocabulary scores are less than 3.74? c) What percent of the scores are between 5.29 and 9.94? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: The height of a giraffe can be described by a normal distribution. Mean, = 204 inches with a standard deviation, = 5.5 inches. Draw a normal curve to represent this information. What percent of giraffes are taller than 215 inches? What percent of giraffes are shorter than 204 inches? What percent of giraffes are taller than 198.5 inches? A height of 209.5 inch corresponds to what percentile of giraffe height? A student states that he got a 640 on both the verbal and math sections of the SAT. The student stated, “I feel stronger in math, so I don’t understand what went wrong.” After researching the College Board’s website the following was found: Math: = 455, = 100 Verbal: = 500, = 85. Does this imply that the math score is actually better or worse? Explain. SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: STANDARD NORMAL CALCULATIONS The standard normal distribution is ________ What if we know that data is distributed normally, but has a mean and standard deviation other than 0 and 1? allows us to change the units so that they can be compared with the standard normal distribution. Given: 1) 2) 3) The standardized value of x is given by: This new value is called the : Ex: Former NBA star Michael Jordan is 78 in. tall, while WNBA player Rebecca Lobo is 76 tall. Men’s heights have a mean of 69 in, and a standard deviation of 2.8 in. Women’s heights have a mean of 63.6 in, and a standard deviation of 2.5 in. Which player is relatively taller? In other words, which player is taller in comparison to their gender? Former NBA player Mugsy Bogues is 63 in. tall. Whose height is more “unusual”, Mugsy’s or Michael’s? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81. How to solve problems involving normal distributions: State: Plan: Do : Conclude : When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards? HOMEWORK Consider babies born in the “normal” range of 37 to 43 weeks of gestation. Extensive data supports the assumption that for such babies born in the U.S., birth weight is normally distributed with a mean of 3432 g, and a standard deviation of 482 g. 1: What is the probability that the birth weight of a randomly chosen baby of this type is less than 4000 g? 2: What is the probability that the birth weight of a randomly chosen baby of this type is more than 3000 g? 3: What is the probability that the birth weight will be between 3000 and 4000 g? 4: What is the probability that the birth weight will be between 2500 and 3500 g? 5: What is the 30% percentile of babies’ weights? What is the interquartile range? SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: SUMMARY/QUESTIONS TO ASK IN CLASS Name _______________________________ AP STATISTICS CHAPTER 2: SUMMARY/QUESTIONS TO ASK IN CLASS