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- WE'RE NOW GOING TO TAKE A LOOK AT AN EXAMPLE
OF LINEAR REGRESSION.
HERE WE'RE GIVEN DATA COMPARING
THE DROP HEIGHT AND BOUNCE HEIGHT OF A BALL.
NOTICE THE FIRST COLUMN IS THE DROP HEIGHT.
THE SECOND COLUMN IS THE BOUNCE HEIGHT.
SO THE FIRST STEP IS TO ANALYZE THE DATA
BY MAKING A SCATTER PLOT WHICH MEANS WE'RE GOING TO PLOT
THESE POINTS ON THE COORDINATE PLANE
WHERE THE FIRST COLUMN WILL BE THE X COORDINATES
AND THE SECOND COLUMN WILL BE THE Y COORDINATES.
BEFORE WE DO THIS WE WANT TO SCALE OUR AXES
TO MAKE SURE IT ACCOMMODATES THE GIVEN X AND Y COORDINATES.
NOTICE THE X COORDINATES RANGE FROM 6 TO 36,
SO LET'S GO AHEAD AND LABEL THIS FROM 0 TO 40
AND THEN BREAK THIS IN HALF THAT WOULD BE 20.
THEN WE'LL GO AHEAD AND CUT THIS IN HALF AS WELL.
THAT WOULD BE 10,
AND THIS WOULD BE 30.
AND NOTICE HOW IT IS LABELED IN INCHES.
AND THEN FOR THE BALANCE HEIGHT,
NOTICE THE RANGE IS FROM 3 TO 18.1,
SO LET'S GO AHEAD AND MARK 20 AT THE TOP.
WE'LL CUT THIS IN HALF. CALL THIS 10, 5,
AND THIS WOULD 15, AGAIN, IN INCHES.
AND NOW WE'LL GO AHEAD AND JUST PLOT THESE POINTS.
SO THE FIRST POINT WOULD HAVE COORDINATES (6, 3),
SO THAT'S GOING TO BE SOMEWHERE IN HERE.
NEXT POINT IS (12, 6.4), MAYBE SOMEWHERE IN HERE.
THE NEXT POINT WOULD BE (18, 8.9),
MAYBE APPROXIMATELY HERE.
NEXT WE HAVE (24, 12.8), SO THAT WOULD BE SOMEWHERE IN HERE.
THEN WE HAVE (30, 14,3).
THE LAST POINT WOULD HAVE COORDINATES (36, 18.1).
NOW, ONCE WE HAVE THE SCATTER PLOT,
WE CAN SEE THE DATA IS BEHAVING IN A LINEAR FASHION
WHICH MEANS THAT A LINEAR EQUATION
WOULD BE A GOOD MODEL FOR THIS DATA.
NOTICE HOW THEY'RE ASKING US TO SKETCH A BEST-FIT LINE
FOR THE DATA, SO WE'LL DO THIS TWO WAYS.
WE'LL FIRST JUST DO IT BY HAND,
AND THEN WE'LL USE THE CALCULATOR AS WELL.
SO IF WE WERE GOING TO SKETCH A LINE TO REPRESENT THIS DATA,
WE'D WANT A LINE THAT PASSES
THROUGH THE MIDDLE OF THESE POINTS
OR MAYBE SOMETHING LIKE THIS.
AND NOW WE'LL GO AHEAD AND USE THE GRAPHING CALCULATOR
TO DETERMINE THE EQUATION OF THIS BEST FIT LINE.
SO OUR DIRECTIONS ARE USE A GRAPHING CALCULATOR
AND PERFORM THE NEW REGRESSION.
ROUND TO THE TENTHS, AND THEN USE THIS EQUATION
TO PREDICT THE BOUNCE HEIGHT IF THE BALL
IS DROPPED FROM 52 INCHES.
SO THE FIRST STEP IS TO ENTER THE DATA INTO THE CALCULATOR.
SO WE'RE GOING TO PRESS THE STAT KEY AND THEN ENTER.
AND NOTICE HOW I HAVE SOME OLD DATA IN HERE,
SO I'M GOING TO GO TO THE TOP OF THE COLUMN,
PRESS CLEAR AND THEN ENTER.
AND IT CLEARS THE ENTIRE COLUMN.
SO WE'LL DO THE SAME FOR L1, CLEAR AND THEN ENTER.
AND NOW WE'LL ENTER THE DATA.
NOW WE'LL ENTER THE SECOND COLUMN WITH THE Y VALUES.
IT IS IMPORTANT THAT WE DOUBLE-CHECK THESE VALUES
BECAUSE IF ONE OF THEM IS OFF THE EQUATION
WILL BE OFF AS WELL.
AND NOW INSTEAD OF JUST DETERMINING THE EQUATION,
WE'RE ALSO GOING TO CREATE THE SCATTER PLOT ON THE CALCULATOR
SO THAT WE CAN GRAPH THE LINEAR EQUATION
RIGHT ON TOP OF THE SCATTER PLOT.
SO THE FIRST TYPE WILL BE TO ADJUST THE WINDOW
JUST LIKE WE LABELED THE AXES.
SO WE'LL PRESS WINDOW.
LET'S CHANGE THE X MINIMUM TO -5 TO MAKE SURE WE SEE THE ORIGIN,
THEN AGAIN THE X MAX WE'LL USE 40.
AND WE'LL SCALE THIS BY 5'S.
AND THEN SAME THING FOR A Y MIN.
WE'LL START AT -5, AND THE Y MAX WE CAN LEAVE AT 20,
THEN WE'LL SCALE THIS BY 5'S AS WELL.
SO NOW IF WE PRESS 2nd Y EQUALS WE'RE GOING TO GO AHEAD
AND TURN THE FIRST STAT PLOT ON,
SO WE'LL PRESS ENTER, HIGHLIGHT ON, AND PRESS ENTER.
AND WITH THESE SETTINGS HERE,
SCATTER PLOT, L1, L2 IN LITTLE BOXES,
SO FROM HERE IF WE PRESS GRAPH, WE CAN SEE OUR SCATTER PLOT,
AND NOW WE WANT TO PERFORM LINEAR REGRESSION
AND THEN GRAPH THAT LINE RIGHT ON TOP OF OUR SCATTER PLOT.
SO WE'RE GOING TO PRESS THE STAT KEY,
RIGHT ARROW ONCE TO CALCULATION,
AND THEN NUMBER FOUR IS FOR LINEAR REGRESSION.
NOTICE THAT "A" IS GOING TO BE THE SLOPE
AND B WILL BE THE Y-INTERCEPT.
SO WE'LL ARROW DOWN TO NUMBER FOUR AND THEN PRESS ENTER
AND THEN ENTER ONE MORE TIME,
AND HERE'S THE EQUATION FOR THE LINE OF BEST FIT.
ONE THING WE HAVE TO BE CAREFUL ABOUT, THOUGH,
IS THEY ARE ASKING US TO ROUND TO THE TENTHS,
SO WE'RE GOING TO ROUND THE SLOPE TO 0.5,
AND WE'LL ROUND THE Y-INTERCEPT TO 0.3.
SO OUR EQUATION IS GOING TO BE Y = 0.5X + 0.3
WHERE X EQUALS THE DROP HEIGHT IN INCHES
AND Y WOULD BE THE BOUNCE HEIGHT IN INCHES.
LET'S GO AHEAD AND ENTER THIS EQUATION TO Y1,
SO WE'LL PRESS Y = AND THEN 0.5X + 0.3.
NOW IF WE PRESS GRAPH WE CAN SEE THAT LINE IS AN EXCELLENT FIT
FOR THE GIVEN DATA.
AND NOW TO ANSWER THE QUESTION,
WE WANT TO KNOW THE BOUNCE HEIGHT
IF THE DROP HEIGHT IS 52 INCHES.
IF WE WANT TO DO THIS BY HAND,
WE WOULD REPLACE X WITH 52 AND THEN SIMPLIFY,
BUT I'LL ALSO SHOW YOU HOW YOU CAN DO THIS ON THE CALCULATOR.
WE COULD USE THE TABLE FEATURE,
OR WE CAN GO BACK TO THE HOME SCREEN, PRESS 2nd MODE, VARS,
RIGHT ARROW, ENTER, ENTER TO SELECT Y1
AND THEN IN PARENTHESES WE'LL TYPE IN 52.
THIS SHOULD REMIND US OF FUNCTION NOTATION,
AND WE CAN SEE HERE WE'RE GOING TO APPROXIMATE
THE BOUNCE HEIGHT TO BE 26.3,
AND THIS WOULD BE INCHES.
AND THAT'LL DO IT FOR THIS EXAMPLE.