LINE OF BEST FIT LINE OF BEST FIT

LINE OF BEST FIT

LINE OF BEST FIT Remember: Line of Best Fit = is the line that best approximates the linear relationship between two data sets (comes closest to all of the dots on the graph) Linear Regression Function = is the equation for the line of best fit Usually there is no single line that passes through all the data points, so you try to find the line that best fits / represents most data.

FINDING THE LINE OF BEST FIT Step 1: using a ruler, place it on the graph to find where the edge of the ruler touches the most points. § Make sure it touches at least 2 points (best if they are far apart) § Try to have approximately half of the data points above the line and half below § Goal is to “center” the line in the data points, match slope of general data trend Step 2: Draw in the line.

FINDING THE LINE OF BEST FIT Step 3: Calculate the slope between two points on your line (best if they are far apart) Step 4: Calculate the y intercept algebraically § by substituting the slope and one point on the line into slope-intercept form of an equation and solve for “b. ” Step 5: Write the equation of the line in slopeintercept form.

PRACTICE PROBLEM… THE OLYMPIC GAMES DISCUS THROW Year 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 Winning throw 134. 2 145. 1 146. 6 151. 4 155. 2 162. 4 165. 6 173. 2 180. 5 184. 9 194. 2 200. 1 212. 5 211. 4 221. 5 218. 7 218. 5 225. 8 213. 7 227. 7 The Olympic games discus throws from 1908 to 1996 are shown on the table. Approximate the best fitting line for these throws let x represent the year with x = 8 corresponding to 1908. Let y represent the winning throw. View scatter plot on handout.

STEP 1 & 2: PLACE YOUR RULER ON THE PAGE AND DRAW A LINE WHERE IT TOUCHES THE MOST POINTS ON THE GRAPH.

STEP 3: FIND THE SLOPE BETWEEN 2 POINTS ON THE LINE. The line went right through the point at 1960 and 1988. The ordered pairs for these points are (60, 194. 2) and (88, 225. 8). m = y 2 – y 1 =8 x 2 – x 1 m = 8 7 = 225. 8 – 194. 2 = 31. 6 = 32 88 – 60 28 28 7

STEP 4: FIND THE Y-INTERCEPT. Substitute the slope and one point into the slope-intercept form of an equation. Slope: 8 / 7 and a point: (88, 225. 8) y = mx + b 225. 8 = 8 / 7 (88) + b 225. 8 = 704 / 7 + b 225. 8 = ≈100. 6 + b -100. 6 125. 2 = b

STEP 5: WRITE IN SLOPE-INTERCEPT FORM. Substitute each value into y = mx + b. The linear regression equation is: y = 8 / 7 x + 125. 2 When you solve these problems, you can get different answers for the line of best fit if you choose different points. But the equations should be close.

TRY IT AGAIN… The circus performs 10 times. Each time they keep data on the number of water bottles and lunch boxes sold. Use the data provided to make a scatter plot. Draw the line of best fit. Write the equation of the line of best fit.

Draw the line of best fit. Find the slope. (20, 67) & (58, 34) 34 – 67 = -33 58 – 20 38 Plug into y = mx + b Is this a strong or weak correlation? 67 = - 33 / 38 (20) + b Does that make it easier or harder to Solve for b place the line of best fit? 67 = -17. 7 + b b = 84. 7 Write in slope-intercept form y = - 33 / 38 x + 84. 7

HISTORY VS. MATH Here is a scatter plot showing the relationship between students who took a History Test and a Math Test. Is there a relationship between the scores? Describe the relationship.

HISTORY VS. MATH Since there is a positive, linear correlation with the data, predict what a student who earned a 75% on their history test earned on their math test. What can I draw that will help me make that prediction?

HISTORY VS. MATH 75% on History Test The line of best fit will help you make a prediction as to what score the student would get on their math test if they earned a 75% on their history test. About 77% What score would he get on the math test?

POPULATION OF GOLDFISH & STAR FISH The following table shows the population between goldfish and star fish at different aquariums. Goldfish 13 18 19 20 27 16 22 24 32 12 Star Fish 30 25 20 15 12 28 18 13 10 29 § Is there a relationship in this data? § What can we draw to see if there is a relationship? § Draw a scatter plot

GOLDFISH & STAR FISH Is there a relationship with this data? What kind of relationship is it? If I have a goldfish population of 15, how many star fish will there be? What can I draw to help me make that prediction?

GOLDFISH & STAR FISH Draw a line of best fit. If I have a goldfish population of 15, how many star fish will there be? There would be about 26 star fish.