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Chapter 3 concepts/objectives Define and describe density curves Measure position using percentiles Measure position using z-scores Describe Normal distributions Describe and apply the 68-95-99.7 Rule Describe the standard Normal distribution Perform Normal calculations 1

2 The 68-95-99.7 Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ. The 68-95-99.7 Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ. σ

3 The Standard Normal Distribution  All Normal distributions are the same if we measure in units of size σ from the mean µ as center. 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 has the standard Normal distribution, N(0,1). 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 has the standard Normal distribution, N(0,1).

4 Normal Calculations Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81. Can you find the same proportion using a different approach? 1 – (0.1056+0.2090) = 1 – 0.3146 = 0.6854

5 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. Solve: 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: State the conclusion in the context. 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. Solve: 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: State the conclusion in the context. How to Solve Problems Involving Normal Distributions Normal Calculations

Chapter 4 concepts/objectives Explanatory and Response Variables Displaying Relationships: Scatterplots Interpreting Scatterplots Measuring Linear Association: Correlation Facts About Correlation 6

The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ from the mean µ as center. 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. has the standard Normal distribution, N(0,1).

Measuring Linear Association A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables The correlation r measures the strength of the linear relationship between two quantitative variables. r is always a number between -1 and 1. r > 0 indicates a positive association. r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 toward -1 or 1. The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

Facts About Correlation 1.Correlation makes no distinction between explanatory and response variables. 2. r has no units and does not change when we change the units of measurement of x, y, or both. 3. Positive r indicates positive association between the variables, and negative r indicates negative association. 4. The correlation r is always a number between -1 and 1. Cautions: Correlation requires that both variables be quantitative. Correlation does not describe curved relationships between variables, no matter how strong the relationship is. Correlation is not resistant. r is strongly affected by a few outlying observations. Correlation is not a complete summary of two-variable data.

10 Regression equation: y = a + bx ^  x is the value of the explanatory variable.  “y-hat” is the predicted value of the response variable for a given value of x ( based on the line of best fit ).  b is the slope, the amount by which y changes for each one-unit increase in x.  a is the intercept, the value of y when x = 0. Chapter 5 -- Regression Line

11 Least Squares Regression Line To predict y, we want the regression line to be as close as possible to the data points in the y (vertical) direction. Least Squares Regression Line (LSRL): The line that minimizes the sum of the squares of the vertical distances of the data points from the line. For LSRL, the constants a (intercept) and b (slope) are calculated and inserted in the regression line. Regression equation: y = a + bx Calculate b from: Calculate a from: where s x and s y are the standard deviations of the two variables x and y, and r is their correlation. Least Squares Regression Line (LSRL): The line that minimizes the sum of the squares of the vertical distances of the data points from the line. For LSRL, the constants a (intercept) and b (slope) are calculated and inserted in the regression line. Regression equation: y = a + bx Calculate b from: Calculate a from: where s x and s y are the standard deviations of the two variables x and y, and r is their correlation. ^

12 An outlier is an observation that lies far away from the other observations. –Outliers in the y direction have large residuals. –Outliers in the x direction are often influential for the least-squares regression line, meaning that the removal of such points would markedly change the equation of the line. Outliers and Influential Points

13 Chapter 6 --Two-Way Table, Example What are the variables described by this two-way table? (Hint: Number of columns?) How many young adults were surveyed? (Hint: It is one of the totals in bottom row.)

14 Chap 6, Marginal Distribution The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution: 1.Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2. Make a graph to display the marginal distribution.

15 Chap. 6 -- Conditional Distribution Marginal distributions tell us nothing about the relationship between two variables. A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions: 1.Select the row(s) or column(s) of interest. 2.Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3.Make a graph to display the conditional distribution. Use a side-by-side bar graph or segmented bar graph to compare distributions. 15

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