5.7 SCATTER PLOTS AND TREND LINES: Scatter Plot: a graph that relates two different sets of data by displaying them as ordered pairs (x, y). Correlation: The relationship/trend found in any given data. Trend line: Line on a scatter plot, drawn near the points, that shows a correlation

Interpolation: The act of estimating a value between two known values. Extrapolation: The act of predicting a value outside of the range of known values. Line of Best Fit: Line that shows the most accurate relationship between two sets of data.

Correlation Coefficient(r): a number from -1 to 1 that tells us how closely the equation models the data. Causation: A change in one quantity causes a change in a second quantity.

GOAL:

SCATTER PLOTS: We must be able to provide domain, range and ordered pairs.

Positive Correlation: CORRELATIONS: In any data we can have three types of correlation: y Positive Correlation: x Our data increases from left to right.

Negative Correlation: y x Our data decreases from left to right.

No Correlation: y x Our data does not have any pattern.

Correlation Coefficient (r):

Whenever we are given data/information/ordered pairs, we must be able to provide certain details: Ex: Make a scatter plot of the data, provide the type of relationship it represents and the approximate weight of a 7-month-old panda. Weight of a Panda Age (months) 1 2 3 4 6 8 10 12 Weight (lbs) 2.5 7.6 12.5 17.1 24.3 37.9 49.2 54.9

To answer the questions on the panda task we must do three procedures: Procedure 1: Create a Scatter Plot Procedure 2: Write an Equation of the Trend of the Line Procedure 3: Estimate the weight of a 7-month-old panda.

Positive Correlation Trend Line Procedure 1: Scatter Plot Age (x) 1 2 3 4 6 8 10 12 Weight(y) 2.5 7.6 12.5 17.1 24.3 37.9 49.2 54.9 Positive Correlation Trend Line 60 50 Weight (lbs) 40 30 20 10 2 4 6 8 10 12 Age (Months)

𝒚 =5.2𝒙 – 3.7 Procedure 2: Write an equation of the Trend Using A(4, 17.1) and B(8, 37.9), points on the positive correlation line, we find the slope Age Weight 1 2.5 2 7.6 3 12.5 4 17.1 6 24.3 8 37.9 10 49.2 12 54.9 m= 𝟑𝟕.𝟗 −𝟏𝟕.𝟏 𝟖−𝟒 = 𝟐𝟎.𝟖 𝟒 = 5.2 Using one of the two points and the point-slope form equation: 𝒚- 𝒚𝟏 = m(𝒙-𝒙𝟏) we get: 𝒚- 17.1 =5.2(𝒙 - 4) 𝒚- 17.1 =5.2𝒙 – 20.8 𝒚 =5.2𝒙 – 20.8 +17.1 𝒚 =5.2𝒙 – 3.7

Thus a 7-month-old panda will weight about 32.7 lbs. Procedure 3: Estimate the weight of the 7-month-old panda Using the found equation of the Trend Line: 𝒚 =5.2𝒙 – 3.7 and letting x = 7 months, we get: 𝒚 =5.2(7) – 3.7 𝒚 =36.4 – 3.7  𝒚 =32.7 Thus a 7-month-old panda will weight about 32.7 lbs.

YOU TRY IT: Use the data below to create a scatter plot, provide the relationship and approximate the daily temperature in January at a latitude of 50o N. Latitude Temp 35 46 33 52 30 67 25 76 43 32 40 37 39 44

Negative Correlation Trend Line Procedure 1: Scatter Plot Latitude (x) Temp (y) 35 46 33 52 30 67 25 76 43 32 40 37 39 44 80 70 60 50 40 Temperature (o F) 30 20 10 Negative Correlation Trend Line 20 25 30 35 40 45 50 Latitude (o N)

𝒚 = -3𝒙+157 Procedure 2: Write an equation of the Trend Using A(30, 67) and B(40, 37), points on the negative correlation line, we find the slope (x) (y) 35 46 33 52 30 67 25 76 43 32 40 37 39 44 m= 𝟑𝟕−𝟔𝟕 𝟒𝟎−𝟑𝟎 = −𝟑𝟎 𝟏𝟎 = - 3 Using one of the two points and the point-slope form equation: 𝒚- 𝒚𝟏 = m(𝒙-𝒙𝟏) we get: 𝒚- 67 = -3 (𝒙 - 30) 𝒚- 𝟔𝟕 =-3𝒙+𝟗𝟎 𝒚 =-3𝒙 + 90 + 67 𝒚 = -3𝒙+157

Thus at latitude of 50o N the temperature will be about 7o F. Procedure 3: Estimate the temperature of the 50oN: Using the found equation of the Trend Line: 𝒚 = -3𝒙+157 and letting x = 50o N of latitude we get: 𝒚 =-3(50)+𝟏𝟓𝟕 𝒚 = –150 + 157  𝒚 =7 Thus at latitude of 50o N the temperature will be about 7o F.

VIDEOS: Scatter Plots https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-slope-intercept/v/fitting-a-line-to-data

CLASSWORK: Page 338-339 Problems: As many as needed to master the concept