The Weather Turbulence Unit 4 – Bivariate Data 12/6/2018 Algebra 1 Institute

Could a city’s elevation above sea level be used to predict the number of cloudy, partly cloudy, or clear days per year a city experiences? 12/6/2018 Algebra 1 Institute

Scatterplot Do you see a pattern in the scatter plot, or does it look like the data points are scattered? How would you describe the relationship between elevation and mean number of clear days for these 14 cities? That is, does the mean number of clear days tend to increase as elevation increases, or does the mean number of clear days tend to decrease as elevation increases? Do you think that a straight line would be a good way to describe the relationship between the mean number of clear days and elevation? Why do you think this? 12/6/2018 Algebra 1 Institute

How Strong is the Pattern? Correlation Coefficient The value of the correlation coefficient ranges between –1 and +1. The greater the absolute value of a correlation coefficient, the stronger the linear relationship. 12/6/2018 Algebra 1 Institute

How Strong is the Pattern? Correlation Coefficient The strongest linear relationship is indicated by a correlation coefficient of –1 or +1. The weakest linear relationship is indicated by a correlation coefficient equal to 0. A positive correlation means that if one variable gets bigger, the other variable tends to get bigger. A negative correlation means that if one variable gets bigger, the other variable tends to get smaller. 12/6/2018 Algebra 1 Institute

12/6/2018 Algebra 1 Institute

Calculate the Correlation Coefficient where is the deviation score and is the sample mean (Software!) 12/6/2018 Algebra 1 Institute

Coefficient of Determination – R2 The coefficient of determination is the square of the correlation coefficient R2 = r2 The correlation coefficient ranges from –1 to +1. So the coefficient of determination ranges from 0 to +1. 12/6/2018 Algebra 1 Institute

Coefficient of Determination – R2 X = independent variable (predictor) Y = dependent variable (response) An R2 of 0 means that X does not explain any of the variation in Y. An R2 of 1 means X explains all the variation in Y. Intermediate values: an R2 between 0 and 1 indicates the extent to which Y is explained by X R2 of 0.10 means that 10 percent of the variation in Y is explainable from X R2 of 0.20 means that 20 percent is explained; and so on. 12/6/2018 Algebra 1 Institute 12/6/2018 Algebra 1 Institute 9

The Least Squares Regression Line ŷ = b0 + b1x Where b0 is an estimate of the intercept b1 is the estimate of the slope b0 and b1 are regression coefficients x is the value of the independent variable, ŷ is the predicted (fitted) value of the dependent variable. 12/6/2018 Algebra 1 Institute

The Least Squares Regression Line Estimates: or where and (Software!) 12/6/2018 Algebra 1 Institute

The Least Squares Regression Line Estimates: 12/6/2018 Algebra 1 Institute

The Weather Turbulence Construct a scatter plot that displays the data for x = elevation above sea level (in feet), and, w = mean number of partly cloudy days per year Based on the scatter plot you constructed, Is there a relationship between elevation and the mean number of partly cloudy days per year? If so, how would you describe the relationship? Explain your reasoning. 12/6/2018 Algebra 1 Institute