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 3 4.60 86 6 3 1
We can pedict on fom: m + b whee m (.60)(6).90 4 b m 86 (.9)3 57..90 + 57. 4 We can pedict on fom: m + b whee m (.90)(4).6 6 b m 3 (.6)(86) 19.6.6 19.6 5 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
7 8.5.68 16% 16%.31.41 34 PR of 65 o 66 9 3
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. 10 11 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 13-15 (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 1.5 13.5, 13.5 14.5, 14.5 15.5. 1 4
) Aveages The Mean is the aithmetic aveage of scoes. Example:, 3, 5, 10 Mean 50/4 Σ n Example: f f 8-10 18 5-7 3 18-4 4 1 um 38 mean 38 4. 9 Σf n 13 14 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
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) 9 + 4 + 0 + 5 38 9.5 4 4 4 nσ ( Σ ) ( 4)[ 138] ( 0) 9.50 n 16 16 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: M5 3.08 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 ) 3 5 15 + 10 5 18 6
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: 65 50 Z 1.50 10 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 50-43.3 6.68%. 0 1 7
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,,.60.60 + 3, 5,.60.60 4 8
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 (0, 3) lope.0 Intecept 3. 0 + 3 3 0 3.6 3 6 9