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**Chapter 3 ~ Descriptive Analysis & Presentation of Bivariate Data**

5 4 3 2 1 6 Weight Height Regression Plot Y = X r = 0.559

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Chapter Goals To be able to present bivariate data in tabular and graphic form To become familiar with the ideas of descriptive presentation To gain an understanding of the distinction between the basic purposes of correlation analysis and regression analysis

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3.1 ~ Bivariate Data Bivariate Data: Consists of the values of two different response variables that are obtained from the same population of interest Three combinations of variable types: 1. Both variables are qualitative (attribute) 2. One variable is qualitative (attribute) and the other is quantitative (numerical) 3. Both variables are quantitative (both numerical)

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**Two Qualitative Variables**

When bivariate data results from two qualitative (attribute or categorical) variables, the data is often arranged on a cross-tabulation or contingency table Example: A survey was conducted to investigate the relationship between preferences for television, radio, or newspaper for national news, and gender. The results are given in the table below:

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Marginal Totals This table may be extended to display the marginal totals (or marginals). The total of the marginal totals is the grand total: Row Totals 760 560 Col. Totals 395 450 475 1320 TV Radio NP Male 280 175 305 Female 115 275 170 Note: Contingency tables often show percentages (relative frequencies). These percentages are based on the entire sample or on the subsample (row or column) classifications.

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**Percentages Based on the Grand Total (Entire Sample)**

The previous contingency table may be converted to percentages of the grand total by dividing each frequency by the grand total and multiplying by 100 For example, 175 becomes 13.3% TV Radio NP Row Totals Male 21.2 13.3 23.1 57.6 Female 8.7 20.8 12.9 42.4 Col. Totals 29.9 34.1 36.0 100.0 175 1320 100 13 3 = æ è ç ö ø ÷ .

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**Percentages Based on Grand Total**

Illustration These same statistics (numerical values describing sample results) can be shown in a (side-by-side) bar graph: 5 10 15 20 25 TV Radio NP Male Female Percentages Based on Grand Total Percent Media

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**Percentages Based on Row (Column) Totals**

The entries in a contingency table may also be expressed as percentages of the row (column) totals by dividing each row (column) entry by that row’s (column’s) total and multiplying by The entries in the contingency table below are expressed as percentages of the column totals: Note: These statistics may also be displayed in a side-by-side bar graph

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**One Qualitative & One Quantitative Variable**

1. When bivariate data results from one qualitative and one quantitative variable, the quantitative values are viewed as separate samples 2. Each set is identified by levels of the qualitative variable 3. Each sample is described using summary statistics, and the results are displayed for side-by-side comparison 4. Statistics for comparison: measures of central tendency, measures of variation, 5-number summary 5. Graphs for comparison: dotplot, boxplot

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Example Example: A random sample of households from three different parts of the country was obtained and their electric bill for June was recorded. The data is given in the table below: The part of the country is a qualitative variable with three levels of response. The electric bill is a quantitative variable. The electric bills may be compared with numerical and graphical techniques.

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**Comparison Using Dotplots**

: Northeast . :..:. .. Midwest : West The electric bills in the Northeast tend to be more spread out than those in the Midwest. The bills in the West tend to be higher than both those in the Northeast and Midwest.

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**Comparison Using Box-and-Whisker Plots**

2 3 4 5 6 7 Electric Bill The Monthly Electric Bill

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**Two Quantitative Variables**

1. Expressed as ordered pairs: (x, y) 2. x: input variable, independent variable y: output variable, dependent variable Scatter Diagram: A plot of all the ordered pairs of bivariate data on a coordinate axis system. The input variable x is plotted on the horizontal axis, and the output variable y is plotted on the vertical axis. Note: Use scales so that the range of the y-values is equal to or slightly less than the range of the x-values. This creates a window that is approximately square.

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Example Example: In a study involving children’s fear related to being hospitalized, the age and the score each child made on the Child Medical Fear Scale (CMFS) are given in the table below: Construct a scatter diagram for this data

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**Child Medical Fear Scale**

Solution age = input variable, CMFS = output variable Child Medical Fear Scale 1 5 4 3 2 9 8 7 6 CMFS Age

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3.2 ~ Linear Correlation Measures the strength of a linear relationship between two variables As x increases, no definite shift in y: no correlation As x increases, a definite shift in y: correlation Positive correlation: x increases, y increases Negative correlation: x increases, y decreases If the ordered pairs follow a straight-line path: linear correlation

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**Example: No Correlation**

As x increases, there is no definite shift in y: 3 2 1 5 4 Output Input

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**Example: Positive Correlation**

As x increases, y also increases: 5 4 3 2 1 6 Output Input

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**Example: Negative Correlation**

As x increases, y decreases: Output Input 5 4 3 2 1 9 8 7 6

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Please Note Perfect positive correlation: all the points lie along a line with positive slope Perfect negative correlation: all the points lie along a line with negative slope If the points lie along a horizontal or vertical line: no correlation If the points exhibit some other nonlinear pattern: no linear relationship, no correlation Need some way to measure correlation

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3.1 ~ Bivariate Data Coefficient of Linear Correlation: r, measures the strength of the linear relationship between two variables Pearson’s Product Moment Formula: Notes: r = +1: perfect positive correlation r = -1 : perfect negative correlation

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**Alternate Formula for r**

SS “sum of squ ares for ( ) x x” = n - å 2 SS “sum of squ ares for ( ) y y” = n - å 2 SS “sum of squ ares for ( ) xy xy” = x y n - å

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Example Example: The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear correlation coefficient:

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**Completing the Calculation for r**

xy x y = - SS ( ) . )( 0. 42 79 7 449 1116 9 47

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**Please Note r is usually rounded to the nearest hundredth**

r close to 0: little or no linear correlation As the magnitude of r increases, towards -1 or +1, there is an increasingly stronger linear correlation between the two variables Method of estimating r based on the scatter diagram. Window should be approximately square. Useful for checking calculations.

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3.3 ~ Linear Regression Regression analysis finds the equation of the line that best describes the relationship between two variables One use of this equation: to make predictions

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**Models or Prediction Equations**

Some examples of various possible relationships: y ^ b x = + 1 a bx cx 2 ( ) log Linear: Quadratic: Exponential: Logarithmic: Note: What would a scatter diagram look like to suggest each relationship?

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**Method of Least Squares**

b x = + 1 y ^ Equation of the best-fitting line: y ^ Predicted value: ( ) )) y b x - = + å 2 1 ^ Least squares criterion: Find the constants b0 and b1 such that the sum is as small as possible

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**Illustration Observed and predicted values of y: y y b x = + ) ( , x y**

1 y ^ ) ( , x y y - ^ y ^ ( , ) x

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**The Line of Best Fit Equation**

The equation is determined by: b0: y-intercept b1: slope Values that satisfy the least squares criterion:

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Example Example: A recent article measured the job satisfaction of subjects with a 14-question survey. The data below represents the job satisfaction scores, y, and the salaries, x, for a sample of similar individuals: 1) Draw a scatter diagram for this data 2) Find the equation of the line of best fit

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Finding b1 & b0 Preliminary calculations needed to find b1 and b0:

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**Line of Best Fit ( ) å y ^ b xy x 118 75 229 5 5174 = SS ( ) . 0. b y**

1 133 5174 234 8 4902 = - × å (0. )( . Equation o f the line of best f it: . 0. x = + 1 49 517 y ^ Solution 1)

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**Job Satisfaction Survey**

Scatter Diagram 21 23 25 27 29 31 33 35 37 12 13 14 15 16 17 18 19 20 22 Job Satisfaction Salary Job Satisfaction Survey Solution 2)

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Please Note Keep at least three extra decimal places while doing the calculations to ensure an accurate answer When rounding off the calculated values of b0 and b1, always keep at least two significant digits in the final answer The slope b1 represents the predicted change in y per unit increase in x The y-intercept is the value of y where the line of best fit intersects the y-axis The line of best fit will always pass through the point

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Making Predictions 1. One of the main purposes for obtaining a regression equation is for making predictions y ^ 2. For a given value of x, we can predict a value of 3. The regression equation should be used to make predictions only about the population from which the sample was drawn 4. The regression equation should be used only to cover the sample domain on the input variable. You can estimate values outside the domain interval, but use caution and use values close to the domain interval. 5. Use current data. A sample taken in 1987 should not be used to make predictions in 1999.

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