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AP Statistics Chapter 2 - The Normal Distribution 2.1 Describing Location in a Distribution Objective: MEASURE position using percentiles INTERPRET cumulative relative frequency graphs MEASURE position using z-scores TRANSFORM data DEFINE and DESCRIBE density curves Measuring Position: Percentiles The pth percentile of a distribution is the value with p percent of the observations less than it. Examples: Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test: 79 77 81 83 80 86 77 90 73 79 83 85 74 83 93 89 78 84 80 82 75 77 67 72 73 Problem: Use the scores on Mr. Pryor’s test to find the percentiles for the for the following students (how did they perform relative to their classmates): a) Jenny, who earned an 86. b) Norman, who earned a 72. c) Katie, who earned a 93. d) the two students who earned scores of 80. A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each class of a frequency distribution. Cumulative Relative Frequency Graphs Age of First 44 Presidents When They Were Inaugurated Age Frequency 40-44 2 45-49 7 50-54 13 55-59 12 60-64 7 65-69 3 Relative frequency Cumulative frequency Cumulative relative frequency 1 Was Barack Obama, who was inaugurated at age 47, unusually young? Was Ronald Reagan, who was inaugurated at age 69, unusually old? Example: Measuring Position: z-Scores Check Your Understanding page 89 A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. To compare data from distributions with different means and standard deviations, we need to find a common scale. We accomplish this by using standard deviation units (z-scores) as our scale. Changing to these units is called standardizing. Standardizing data shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation. z score datavalue mean st .dev. or z x Standardizing does not change the shape of the distribution. It changes the center (shifts it to zero) and the spread by making the standard deviation one. Example: #11 page 106 2 Transforming Data Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution. Effect of Adding (or Subracting) a Constant Adding the same number a (either positive, zero, or negative) to each observation: • adds a to measures of center and location (mean, median, quartiles, percentiles), but • Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation). Effect of Multiplying (or Dividing) by a Constant Example: Remember: Exploring Quantitative Data Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): • multiplies (divides) measures of center and location by b • multiplies (divides) measures of spread by |b|, but • does not change the shape of the distribution page 107 # 20 To describe a distribution: - Make a graph - Look for overall patterns (shape, center, and spread) and outliers - Calculate a numerical summary to describe the center (mean, median) and spread (minimum, maximum, Q1, Q3, range, IQR, standard deviation) In addition to the above distributions sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. Density Curves A density curve describes the overall pattern of a distribution o Is always on or above the horizontal axis o Has exactly 1 underneath it o The area under the curve and above any range of values is the proportion of all observations The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars. edian of a density curve is the equal areas point, the point that divides the are under the curve in half Mean of a density curve is the balance point, at which the curve would balance if made of solid 3 Median and Mean of a Density Curve Example: material. Use the figure shown to answer the following questions. 1. Explain why this is a legitimate density curve. 2. About what proportion of observations lie between 7 and 8? 3. Mark the approximate location of the median. 4. Mark the approximate location of the mean. Explain why the mean and median have the relationship that they do in this case. Page 108 #28,30 Examples: Page 108 #32 Assignment page 105-109 #1-31 odd, 33-38 all 4 2.2 Normal Distribution Objectives: DESCRIBE and APPLY the 68-95-99.7 Rule DESCRIBE the standard Normal Distribution PERFORM Normal distribution calculations ASSESS Normality Normal Distributions N(μ,) The 68-95-99.7 Rule All Normal curves have the same overall shape: symmetric, single-peaked, bell shaped. A Normal distribution is described by a Normal density curve. A Normal distribution can be fully described by two parameters, its mean μ and standard deviation σ The mean, µ, of a Normal distribution is at the center of the symmetric Normal curve and is the same as the median. The standard deviation σ controls the spread of a Normal curve. Curves with larger standard deviations are more spread out. The standard deviation, σ, is the distance from the center to the change-of-curvature points on either side. A short-cut notation for the normal distribution in N(μ,). All normal curves obey the 68-95-99.7% (Empirical) Rule. This rule tells us that in a normal distribution approximately 68% of the data values fall within one standard deviation (1σ) of the mean, 95% of the values fall within 2σ of the mean, and 99.7% (almost all) of the values fall within 3σ of the mean. Application of the 68-95-99.7 Rule Distribution of the heights of young women aged 18 to 24 What is the mean μ? What is the ? What is the height range for 95% of young women? What is the percentile for 64.5 in.? What is the percentile for 59.5 in.? What is the percentile for 67 in.? What is the percentile for 72 in.? 5 The Standard Normal Distribution The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable Z-Score Table z x has the standard Normal distribution, N(0,1). Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table. Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Example Check Your Understanding page 119 4-Step Process How to Solve Problems Involving Normal Distributions State: Express the problem in terms of the observed variable x. Plan: Draw a picture of the distribution and shade the area of interest under the curve. Do: Perform calculations. • Standardize x to restate the problem in terms of a standard Normal variable z. • Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve. Conclude: Write your conclusion in the context of the problem. 6 Normal calculations Example: a. Women’s heights are approximately normal with N(64.5, 2.5). What proportion of all young women are less than 68 inches tall? On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When tiger hits his driver, the distance the balls travels follows a Normal distribution with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least 290 yards? What percent of Tiger’s drives travel between 305 and 325? 7 High levels of cholesterol in the blood increase the risk of heart disease. For 14 year old boys, the Using Table A in distribution of blood cholesterol is approximately Normal with mean µ = 170 milligrams of Reverse Example: cholesterol per deciliter of blood (mg/dl) and standard deviation σ = 30 mg/dl. What is the first quartile off the distribution of blood cholesterol? z-scores on the calculator Assessing Normality Technology Corner page 123 Normal Probability Plots Plot the data. • Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped. Check whether the data follow the 68-95-99.7 rule. • Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution. If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot. Example page 127 Technology Corner page 128 Assignment 2.2 #41-65 odd, 68-74 8 Summary 9