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

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The 68-95-99.7 Rule

The 68-95-99.7 RuleIn 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

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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).

μxz -

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Normal CalculationsFind 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

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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

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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).

μxz -

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.

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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

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Least Squares Regression LineTo 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 sx and sy are the standard deviations of the two variables x and y, and r is their correlation.

^

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• 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

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Chapter 6 --Two-Way Table, Example

Young adults by gender and chance of getting rich

Female Male Total

Almost no chance 96 98 194

Some chance, but probably not 426 286 712

A 50-50 chance 696 720 1416

A good chance 663 758 1421

Almost certain 486 597 1083

Total 2367 2459 4826

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.)

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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.

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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.

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