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%O =#DBivariate Data eBivariate 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).&fX ]E }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. ~z  TF 2This table may be extended to display the marginal totals (or marginals). The total of the marginal totals is the grand total. Contingency tables often show percentages (relative frequencies). These percentages are based on the entire sample or on the subsample (row or column) classifications. d3* , ,=  %G 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% &4H  I "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. These statistics may also be displayed in a side-by-side bar graph. &(jJ  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.       e<#&; =K 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.&L  N  O @Example: positive correlation. As x increases, y also increases.RA P <Example: negative correlation. As x increases, y decreases. R=  R Coefficient of linear correlation: r, measures the strength of the linear relationship between two variables. Pearson s product moment formula: Note: 1. 2. r = +1: perfect positive correlation 3. r = -1 : perfect negative correlation !K  ''S Alternate formula for r: 6T 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. R?-KU #To complete the calculation for r: ,$ X Models or prediction equations: Some examples of various possible relationships. Linear: Quadratic: Exponential: Logarithmic: Note: What would a scatter diagram look like to suggest each relationship?ZeGY Method of least squares: Equation of the best-fitting line: Predicted value: Least squares criterion: Find the constants b0 and b1 such that the sum is as small as possible.(  .Z $Observed and predicted values of y: 6%![ The equation of the line of best fit: Determined by b0: slope b1: y-intercept Values that satisfy the least squares criterion: $   6\  0Example: 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. R1]!  ^" Finding b1 and b0: z_# Scatter diagram: &M  ` ̙33` 3` 3333f` 999MMM` f` f3` 3>?" dd@,|?" dd@   " @ `"  n?" dd@   @@``PR    @ ` ` p>> ;(    Z vgֳgֳ ?P  T Click to edit Master title style! !:  Td vgֳgֳ ?  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S   ` vgֳgֳ ?``  =*   `$!vgֳgֳ ?`   ?*   `!vgֳgֳ ?`   ?*H  0޽h? ?  0 @0(  H  0j ? ̙33d 4,0(    T&v1 ?    =*   Td&v1 ? @   ?*   Z&v1 ?   =*   Z$'v1 ? @  ?* H  0j ? ̙33  % 80P(     fT0gֳgֳ ?P      f0gֳgֳ ?   H  0޽h ?   % NF`(     f45gֳgֳ ? v   NA ??a &N3 X $ 00H  0޽h ?   % NFp(     f3gֳgֳ ?    NA ??zgX $ 00H  0޽h ?   % ,$ ( p z     fT3gֳgֳ ?     NA ??M gX $ 00   NA ??_I X $ 00H  0޽h ?   % $B(  $ $0 NA  ? ?h  $ 00  $ N2gֳgֳ? nThese same statistics (numerical values describing sample results) can be shown in a (side-by-side) bar graph.o 2oH $ 0޽h ?   % NF((  ( (  f42gֳgֳ ?   ( NA  ??Sg X  $ 00H ( 0޽h ?   % vn<(  < <  f5gֳgֳ ?   H < 0޽h ?   % NF@(  @ @  ft7gֳgֳ ?   @ NA ??Wi ] X $ 00H @ 0޽h ?   % D<D(  D D H48A  ?-   D N8gֳgֳ?@ z>Scatter diagram: age = input variable, CMFS = output variable ?0H D 0޽h ?   % umL(  L L HT9A  ?  ) L N9gֳgֳ? KExample: no correlation. As x increases, there is no definite shift in y.DL*H L 0޽h ? 8  % Pd(  Pj P BA ?{l P  f8gֳgֳ ?  H P 0޽h ? 8  % Td(  Tj T BA? T  f:gֳgֳ ?  H T 0޽h ?   % 2* \(  \ \  f$vgֳgֳ ?    \ NA ??@ ? X $ 00 \ NA ??=!SwX $ 00H \ 0޽h ? >  % `j(  ` `  ft:gֳgֳ ?   ` NA ??ZsX $ 00 ` NA ??W X $ 00 ` NA ??5  i X $ 00 ` NA ?? R X $ 00H ` 0޽h ? ^   %    d (  d d  f:gֳgֳ ?   d NA  ??fX  $ 00 d NA "??NRX "$ 00 d NA $?? q lX $$ 00 d NA &??q KX &$ 00 d NA (??m,wX ($ 00 d NA *??j4wX *$ 00  d NA ,??X ,$ 00  d NA .?? v X .$ 00  d NA 0?? X 0$ 00  d NA 2??,X 2$ 00  d NA 4??dX 4$ 00H d 0޽h ? >  % 0hj(  h h  f4;gֳgֳ ?   h NA 6??EX 6$ 00 h NA! 8??  /X 8$ 00 h NA" :??:S X :$ 00 h NA# <??3 { X <$ 00H h 0޽h ? L  % @tx(  t t  fT%gֳgֳ ?   t NA$ >??g X >$ 00 t NA% @??g X @$ 00 t NA& B??gY  X B$ 00 t NA' D??` g B X D$ 00H t 0޽h ? `  % Px(  x x  f&gֳgֳ ?   x NA( F?? X F$ 00 x NA) H??wX H$ 00 x NA* J?? [ X J$ 00H x 0޽h ?   % `|H(  | |  ft&gֳgֳ ?   | NA+ L??eX L$ 00jB | B1?//jB | B1?D/pB | H1?=  | NA, N?? k . X N$ 00 | NA, P??% k X P$ 00  | NA- R?? \ X R$ 00  | NA. T?? R W X T$ 00vB  | N1?g@ @ $  | NA/ V??IX V$ 00  | NA0 X??HDX X$ 00 | NA1 Z?? X Z$ 00pB | H1?Z \pB |@ H1? pB | H1?`  pB |@ H1?V RVpB | H1?TYY$pB | H1?   / | NA \?? P X \$ 00 | NA2 ^?? W X ^$ 00H | 0޽h ?   % ,$p(     f&gֳgֳ ?    NA3 `??'Y q X `$ 00  NA4 b??=  ' X b$ 00H  0޽h ?   % TL(     f'gֳgֳ ?     NA5 d??Z{X d$ 00H  0޽h ? "   %    N (   0 NA6 f? ? h f$ 00 8  NT(gֳgֳ? 2Preliminary calculations needed to find b1 and b0:l3 2(    NA7 h??#IX h$ 00  NA8 j??w {X j$ 00  NA9 l??j MDX l$ 00  NA: n??pX n$ 00  NA; p?? NX p$ 00   NA< r??  X r$ 00   NA= t?? XX t$ 00   NA> v?? X v$ 00H  0޽h ?   % H(     f(gֳgֳ ?    NA? x??p-X x$ 00  NA@ z??PX z$ 00  NAA |??` X |$ 00  NAB ~??t  v X ~$ 00  NAC ??S  =X $ 00H  0޽h ?   % @8(    NAD ??tX $ 00   f)gֳgֳ ?  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