# Scatterplots

# Interpreting Scatterplots

Direction: Positive or Negative

Form: Linear or Non-linear

Strength: Weak, Moderate or Strong

Example

Positive, linear, strong relationship between horsepower and weight in tons.

# Correlation Coefficient (r)

# Least Squares Regression Line

A regression line is a line that describes

**how y changes as x changes**Can be used to

**predict**the value of y for a given value of xCalled the

**Least Squares**regression line because it make the**smallest sum of squares**LSRL will always run through the point (mean of x, mean of y)

Formulas (hat = predicted)

**Remember to note what x and y are**Calculation

- Input data

STAT➡️CALC➡️ 4:LinReg(ax+b)

LinReg(ax+b) L1, L2

Catalog (2ND + 0) ➡️ DiagnosticOn

Do LinReg again to display r

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# Coefficient of Determination

R^2=r^2

Coefficient of Determination = (Correlation Coefficient)^2

**Percent**of the**change in y**that is explained by the change by the**change in x**/**least squares regression line**From the previous example, 41.9% of the change in y can be explained by the change in x

# Residuals (≈error)

Residuals = observed/actual y - predicted y

Resid = y - y hat

Resid < 0: Overpredicted

Resid > 0: Underpredicted

Residual Plot

- no pattern = good fit

- Example

X | 3 | 1 | 1.5 | 6 | 2 |
---|---|---|---|---|---|

Y | 10 | 3 | 14 | 15 | 6 |

Y hat | 10.1 | 6.76 | 7.60 | 15.11 | 8.43 |

Residuals | -0.1 | -3.76 | 6.4 | -0.11 | -2.43 |

Calculator

- Type the regression equation in L3 (y hat)

L4 = L2 - L3

Graph L1, L4 (Residuals)

# Examples

- At the summer school, one of Sarah's teachers told her that you can determine air temperature from the number of cricket chirps

What is the explanatory variable, and what it the response variable

Explanatory/independent variable: cricket chirps

Response/dependent variable: air temperature

To determine a formula, Sarah collected data on temperature and number of chirps per minute on 14 occasions. She entered the data into her calculator and did 2-Var Stats. Here are some results. Use this information to find the equation of the least-squares regression line

Xbar | 165.8 |
---|---|

Sx | 32.0 |

Ybar | 76.83 |

Sy | 9.23 |

r | 0.361 |

b= r * Sy / Sx = 0.104

Ybar = a + b*Xbar

a = Ybar - b*Xbar = 59.57

Yhat = 59.57 + 0.14 * x

Where y = air temperature, and x = cricket chirps

One of Sarah's data points was recorded on a particularly hot day (95F). She counted 2432 cricket chirps in one minute. What is the residual for this data point?

- Residual = Y - Yhat = 95 - (59.57 + 0.104 * 2432) = -217.498