The data set healthy breakfast contains the ratings of 77 cereals and the number of grams of sugar contained in each searving. A simple linear regression model considering Sugar as the explanatory variable and Rating as the response variable produced the following regression line. Rating =59.3-2.40*sugars R2 =.557..What is the value of slope, the value of coefficient of determination, the value of the correlation coefficient. What is the correct interpretation of the slope and the coerrelation coefficent.
To answer your question, let's break down the information you provided and explain the different components:
1. The Regression Line:
The regression line represents the relationship between the explanatory variable (sugars) and the response variable (Rating) in a simple linear regression model. In this case, the equation of the regression line is Rating = 59.3 - 2.40 * sugars.
2. The Slope:
From the regression line equation, we can determine the slope. In this case, the slope is -2.40. The slope tells us how the response variable (Rating) changes for each unit change in the explanatory variable (sugars). In other words, for every additional gram of sugar per serving, the rating of the cereal decreases by 2.40 units, on average.
3. The Coefficient of Determination (R^2):
The coefficient of determination (R^2) is a statistical measure that indicates how well the regression line fits the data. In this case, R^2 is 0.557, which means that approximately 55.7% of the variation in the Rating can be explained by the variation in the sugar content.
4. The Correlation Coefficient:
The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient isn't provided directly, but we can calculate it using the square root of R^2. So, the correlation coefficient would be approximately sqrt(0.557) = 0.746. The correlation coefficient ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. Therefore, in this case, there is a strong negative linear relationship between sugar content and cereal rating.
In summary, the interpretation of the slope is that for each additional gram of sugar per serving, the rating of the cereal decreases by 2.40 units, on average. The coefficient of determination (R^2 = 0.557) indicates that approximately 55.7% of the variation in the cereal rating can be explained by the variation in sugar content. The correlation coefficient (approximately 0.746) shows a strong negative linear relationship between sugar content and cereal rating.