Elementary Statistics and Inference. Elementary Statistics and Inference. 11. Regression (cont.) 22S:025 or 7P:025. Lecture 14.

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1 Elementay tatistics and Infeence :05 o 7P:05 Lectue 14 1 Elementay tatistics and Infeence :05 o 7P:05 Chapte 10 (cont.) D. Two Regession Lines uppose two vaiables, and ae obtained on 100 students, with summay statistics: Vaiable Mean D

2 We can pedict on fom: m + b whee m (.60)(6).90 4 b m 86 (.9) We can pedict on fom: m + b whee m (.90)(4).6 6 b m 3 (.6)(86) The two diffeent egession lines can be used to estimate eithe fom o fom. They ae diffeent lines. If.60 and student has PR of 75 on, what is expected PR on? Z ( ) Z Z (.60)(.68).41 6

3 % 16% PR of 65 o

4 E. Review 1) Histogams A ba gaph that displays the shape of a goup of scoes. Fequency Polygon is a boken line gaph connecting the midpoints of the bas of a histogam. A moothed Fequency Polygon descibes the estimated shape of a lage numbe of scoes fom a population of scoes A density distibution has bas that epesent the pecentage of scoes pe scoe unit within an inteval. The height of the ba is detemined by: % of scoesint eval ht width of the int eval Example: A cetain scoe inteval (3 units wide) has 0% of the scoes in a distibution. The height of the ba fo the histogam fo this inteval would be ht 0/3 6.67%. This means about 6.67% of the scoes occu between each of the intevals , ,

5 ) Aveages The Mean is the aithmetic aveage of scoes. Example:, 3, 5, 10 Mean 50/4 Σ n Example: f f um 38 mean Σf n Mean is the point of balance in a distibution. Median is the point on the scoe scale below which 50% of the scoes fall. It is closest to evey othe scoe value. Popeties of the Mean ( ): Σ( ) 0 M + C C + M C C 15 5

6 1) Vaiance The vaiance is aveage of the squaed deviations fom the mean. Σ( ) n Example:, 3, 5, 10 5 ( 5) + (3 5) + (5 5) + (10 5) nσ ( Σ ) ( 4)[ 138] ( 0) 9.50 n The tandad Deviation () is an index in scoe scale units that descibes the distance a scoe is fom aveage in D units and is the oot of. Example: M A scoe of 8.08 is one D unit above mean. 17 Popeties of Vaiance: + (not affected by adding a constant) C C C o C C Example: If 5 ( 3 )

7 4) Nomal Distibution (see table page A104) The distibution is symmetic about the mean, and is descibed by a mathematical model. When scoes ae nomally distibuted, ib t d one can find the pecent of scoes geate than/less than a scoe, when the scoe is conveted to a standad scoe. mean Z D 19 Example: uppose scoes ae nomally distibuted with Mean50 and D 10. The pecent of scoes between the mean and 65 is: Z Fom the table (page A104), the pecentage of scoes between Z ±1.50 is 86.64, so 43.3% ae between the mean and 1.5 since 50% of the scoes ae above the mean, the pecentage geate than Z 1.50 is %

8 5) Coelation is a way to epot the association between two linealy elated vaiables. The index anges between -1 +1, and a value of 0 means no association between the vaiables. a) Computation of nσ ( Σ )( Σ ) ΣZ Z n( Σ ) ( Σ ) n( Σ ) ( Σ ) n Σxy n [ ][ ] Σ( )( ) n 3 If +1 the vaiables ae pefectly positive associated o pefect diect associated. If -1 the vaiable ae pefectly negatively associated. Linea tansfomations of C,, if C > 0 C,, if C < 0 Example:.60,, , 5,

9 6) Regession A linea egession line ( m + b ) is a line that is the closest distance fom each point in the scatte diagam to the line. m slope of the egession line. m the change in fo each unit change in. b m intecept of whee line cosses the vetical axis. 5 Example: If (0, 3) lope.0 Intecept