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1 ES9 Chapter 3 ~ Descriptive Analysis & Presentation of Bivariate Data

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2 ES9 Chapter Goals To be able to present bivariate data in tabular and graphic form To gain an understanding of the distinction between the basic purposes of correlation analysis and regression analysis To become familiar with the ideas of descriptive presentation

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3 ES9 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) 3.1 ~ Bivariate Data Bivariate Data: Consists of the values of two different response variables that are obtained from the same population of interest

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4 ES9 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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5 ES9 Row Totals 760 560 Col. Totals3954504751320 TVRadioNP Male280175305 Female115275170 Marginal Totals This table may be extended to display the marginal totals (or marginals). The total of the marginal totals is the grand total: 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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6 ES9 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 Percentages Based on the Grand Total (Entire Sample) –For example, 175 becomes 13.3% TVRadioNPRow Totals Male21.213.323.157.6 Female8.720.812.942.4 Col. Totals29.934.136.0100.0 175 1320 100133 .

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7 ES9 These same statistics (numerical values describing sample results) can be shown in a (side-by-side) bar graph: Illustration 0 5 10 15 20 25 TVRadioNP Male Female Percentages Based on Grand Total Percent Media

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8 ES9 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 100. 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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9 ES9 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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10 ES9 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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11 ES9.. :...... ---+---------+---------+---------+---------+---------+--- Northeast. :..:... ---+---------+---------+---------+---------+---------+--- Midwest...... :.. ---+---------+---------+---------+---------+---------+--- West 24.0 32.0 40.0 48.0 56.0 64.0 Comparison Using Dotplots 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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12 ES9 Comparison Using Box-and-Whisker Plots NortheastMidwestWest 20 30 40 50 60 70 Electric Bill The Monthly Electric Bill

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13 ES9 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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14 ES9 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: Example Construct a scatter diagram for this data

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15 ES9 age = input variable, CMFS = output variable Solution Child Medical Fear Scale 1514131211109876 50 40 30 20 10 CMFS Age

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16 ES9 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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17 ES9 As x increases, there is no definite shift in y: Example: No Correlation

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18 ES9 As x increases, y also increases: Example: Positive Correlation

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19 ES9 As x increases, y decreases: Example: Negative Correlation

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20 ES9 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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21 ES9 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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22 ES9 Alternate Formula for r SS“sum of squares for()xx” x x n 2 2 SS“sum of squares for()yy” y y n 2 2 SS“sum of squares for()xyxy” xy xy n

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23 ES9 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: Example

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24 ES9 Completing the Calculation for r r xy xy SS () ()(). (.)(.) 0. 4279 744911169 47

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25 ES9 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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26 ES9 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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27 ES9 Models or Prediction Equations Some examples of various possible relationships: Note:What would a scatter diagram look like to suggest each relationship? y ^ bbx 01 yabxcx 2 ^ y()ab x ^ ylogax b ^ Linear: Quadratic: Exponential: Logarithmic:

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28 ES9 Method of Least Squares y ^ Predicted value: ()(())yybbx 2 01 2 y ^ Least squares criterion: –Find the constants b 0 and b 1 such that the sum is as small as possible bbx 01 y ^ Equation of the best-fitting line:

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29 ES9 bbx 01 y ^ Illustration Observed and predicted values of y: y y ^ y ^ (,)x y ^ )(,xy y

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30 ES9 The Line of Best Fit Equation The equation is determined by: b 0 : y-intercept b 1 : slope Values that satisfy the least squares criterion:

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31 ES9 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 Example

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32 ES9 Preliminary calculations needed to find b 1 and b 0 : Finding b 1 & b 0

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33 ES9 Line of Best Fit b xy x 1 11875 2295 5174 SS () ().. 0. b ybx n 0 1 1335174234 8 14902 (0.)(). Equation of the line of best fit:.0.x 149517 y ^ Solution 1)

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34 ES9 Scatter Diagram 212325272931333537 12 13 14 15 16 17 18 19 20 21 22 Job Satisfaction Salary Job Satisfaction Survey Solution 2)

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35 ES9 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 b 0 and b 1, always keep at least two significant digits in the final answer The slope b 1 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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36 ES9 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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