Quantifying Predictability Common Core Algebra 1 Homework

But how do we know how well a regression model fits the data? How do we compare different regression models and choose the best one? This is where the correlation coefficient comes in. The correlation coefficient, denoted by r, is a number between -1 and 1 that measures how closely the data points follow a straight line. The closer r is to 1 or -1, the stronger the linear relationship between the variables. The closer r is to 0, the weaker the linear relationship between the variables.

Quantifying Predictability Common Core Algebra 1 Homework

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The correlation coefficient can also be used to quantify the predictability of a regression model. The predictability is the proportion of variation in one variable that can be explained by another variable using a regression model. For example, if we have a correlation coefficient of 0.8 between height and age, that means that 64% of the variation in height can be predicted by age using a linear regression model. The formula for calculating the predictability is:

predictability = r

The higher the predictability, the better the regression model fits the data and makes accurate predictions.

To find the correlation coefficient and the predictability of a regression model, we can use graphing calculator technology or online tools. For example, we can use this website to enter our data and get the correlation coefficient and the equation of the regression line. We can also see a scatterplot of our data and how well it follows a straight line.

Lets look at an example. Suppose we have collected data on the number of hours students study for a test and their test scores. We want to use a linear regression model to predict test scores based on study hours. Here is our data:

Study HoursTest Score

265

475

685

895

10100

We enter our data into the website and get the following results:

We can see that the correlation coefficient is 0.986, which is very close to 1. This means that there is a strong positive linear relationship between study hours and test scores. The predictability is 0.986, which is about 0.972. This means that 97.2% of the variation in test scores can be explained by study hours using a linear regression model.

The equation of the regression line is y = 4x + 55, where y is the test score and x is the study hours. We can use this equation to make predictions for any given value of x. For example, if we want to predict the test score for someone who studies for 7 hours, we plug in x = 7 into the equation and get y = 4(7) + 55 = 83.

In conclusion, we have learned how to use the correlation coefficient to quantify the predictability of regression models. We have also learned how to find and interpret the correlation coefficient and the predictability using graphing calculator technology or online tools. c481cea774