Mrs. Venneman



2.2 Density Curves and Normal DistributionsLearning Objectives- Estimate the relative locations of the median and mean on a density curve.- Use the 68–95–99.7 rule to estimate areas (proportions of values) in a Normal distribution.-Use Table A or technology to find (i) the proportion of z-values in a specified interval, or (ii) a z-score from a percentile in the standard Normal distribution.-Use Table A or technology to find (i) the proportion of values in a specified interval, or (ii) 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.Density CurveA density curve is a curve that: A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.Identifying Mean and Median on Density Curves254032004000Mean: The mean is the balancing point, at which the curve would balance if made of solid material.39814525273000Median: The median is the equal areas point, the point that divides the area under the curve in half.Describing Density CurvesA density curve is an idealized description of a distribution of data.For actual data: For a density curve: Normal DistributionsOne particularly important class of density curves are the Normal curves, which describe Normal distributions.38506402667000All Normal curves have the same shape: symmetric, single-peaked, and bell-shaped.Any particular Normal distribution is completely specified by two numbers: its mean ? and standard deviation σ. The mean of a Normal distribution is the center of the symmetric Normal curve. The standard deviation is the distance from the center to the change-of-curvature points on either side.We abbreviate the Normal distribution with mean ? and standard deviation σ as N(?,σ).3303270-29718000Example: One source says that the heights of 11-year-old females are approximately Normally distributed with a mean of 59 inches and a standard deviation of 3 inches. Label the distribution with the mean and the points one, two, and three standard deviations from the mean. 193167046101000Example: Here is a dotplot of Kobe Bryant’s point totals for each of the 82 games in the 2008-2009 regular season. The mean of this distribution is 26.8 with a standard deviation of 8.6 points. In what percentage of games did he score within one standard deviation of his mean? …within two standard deviations?3467735850900068-95-99.7 Rule (Empirical Rule)Example: 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?Example: The distribution of blood glucose levels (after 4 hours of fasting and measured in mg/dL) is approximately Normal and the middle 95% of scores are between 70 and 110. What are the mean and standard of this distribution?Calculate the percent of scores that are above 80? Homework: pg. 128-129 # 33-45 odd Standard Normal DistributionFinding Proportions or Percentages using Standard Normal DistributionUsing Table AThe standard Normal 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.Find z-score on left and top, then see where they meet to find the area to the left of that value. Using a Graphing CalculatorGo to distribution menu: 2nd VARSChoose 2:normalcdfEnter values Lower: Upper:220599014478000 ???? ????Hit ENTER5058410-226441000Example: Find the proportion of observations from the standard Normal distribution that are: (a) less than 0.56(b) greater than –1.14 (c) greater than 3.79(d) between 0.46 and 1.84Finding a z-score when you know the proportion or percentageUsing Table ALook inside the table for the closest proportion or percentage, then look left and up to find the z-score2015490-19939000Using a Graphing CalculatorGo do distribution menu: 2nd VARSChoose 3:invNormEnter values Area: ????212979011938000 ????Hit ENTERExample: In the standard Normal distribution, 92% of the observations are less than what value?Tracy fasted for 4 hours and had her blood glucose checked. Her level was at the 85th percentile. How many standard deviations above the mean is that? What percent of a Normal distribution is within .6745 standard deviations of the mean? What do we call the locations at roughly z=-.6745 and at roughly z=.6745?Homework: pg. 129-130 #47, 49, 51, 52Normal Distribution CalculationsFinding Areas/Proportions/Percentages in a Normal DistributionState: State the distribution and the values of interest. The distribution of __ is approximately Normally distributed with a mean of __ and a standard deviation of __. State the question.Plan: Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and boundary value(s) clearly identified.Do: Perform calculations and show your work! Do one of the following: (i) Compute a z-score for each boundary value and use Table A or technology to find the desired area under the standard Normal curve; or (ii) use the normalcdf command and label each of the inputs.Conclude: Answer the question in context.Finding Values from Areas/Proportions/Percentages in a Normal DistributionState: State the distribution and the values of interest. The distribution of __ is approximately Normally distributed with a mean of __ and a standard deviation of __. State the question.Plan: Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and unknown boundary value(s) clearly identified.Do: Perform calculations—show your work! Do one of the following: (i) Use Table A or technology to find the value of z with the indicated area under the standard Normal curve, then “unstandardize” to transform back to the original distribution; or (ii) Use the invNorm command and label each of the inputs.Conclude: Answer the question in context.Example: 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 approximately 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 below 120 mph?(b) What percent of Rafael Nadal’s first serves are between 100 and 110 mph? (c) The fastest 30% of Nadal’s first serves go at least what speed?(d) What is the IQR for the distribution of Nadal’s first serve speeds? (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? Homework: pg. 130-131 #53, 57, 59, 612.2 Assessing Normality To assess Normality: 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.Examine a Normal Probability Plot.Normal Probability PlotTo make: Enter data into graphing calculator and construct a Normal probability plot (bottom right graph in statplot).To interpret: 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: 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) The mean for this set is 15.825 cubic feet and the standard deviation is 1.217 cubic feet. Decide whether you think this data has an approximately Normal distribution.12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.315.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.416.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4 Homework: pg. 131-133 # 63, 65, 66, 67, 69-74Chapter 2 Learning ObjectivesSectionRelated Example on Page(s)Relevant Chapter Review Exercise(s)Can I do this?Find and interpret the percentile of an individual value within a distribution of data.2.186R2.1 Estimate percentiles and individual values using a cumulative relative frequency graph.2.187, 88R2.2Find and interpret the standardized score (z-score) of an individual value within a distribution of data.2.190, 91R2.1Describe the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.2.193, 94, 95R2.3Estimate the relative locations of the median and mean on a density curve.2.2Discussion on 106–107R2.4Use the 68–95–99.7 rule to estimate areas (proportions of values) in a Normal distribution.2.2111R2.5Use Table A or technology to find (i) the proportion of z-values in a specified interval, or (ii) a z-score from a percentile in the standard Normal distribution.2.2114, 115, Discussion on 116R2.6Use Table A or technology to find (i) the proportion of values in a specified interval, or (ii) the value that corresponds to a given percentile in any Normal distribution.2.2118, 119, 120R2.7, R2.8, R2.9Determine whether a distribution of data is approximately Normal from graphical and numerical evidence.2.2122, 123, 124R2.10, R2.11 ................
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