Final answer:
The least-squares estimate of the mean diameter to height ratio is calculated using the slope, β1, from the regression equation y = mx + b. The slope determined as m ≈ 0.09 and the y-intercept as b ≈ 35.25 help us predict weight or diameter from a given height. The coefficient of determination is used to understand the proportion of variability explained by the linear relationship in a correlation.
Step-by-step explanation:
To calculate the least-squares estimate of the mean diameter to height ratio for the given data, we should apply a line of best fit using the least-squares method. The formula for a line is y = mx + b, where m represents the slope, and b the y-intercept. The slope of this line, β1, is what we are looking for. It's the ratio of the mean diameter (y) to the height (x).
To find this slope, we take the sum of the product of each x and its corresponding y values and divide it by the sum of the squares of the x values. However, from the information provided, we know that the slope is already computed as m ≈ 0.09. This slope indicates the change in the mean diameter for each unit change in height. Following that, we calculate the y-intercept using the formula b = Σy - m(Σx), where Σy is the sum of y-values, and Σx is the sum of x-values. Our provided sums give us b ≈ 35.25. So, the regression equation is y = 0.09x + 35.25.
Applying the regression equation to a specific height, say 68 inches, to predict weight (assuming this question wants weight instead of diameter), we substitute x with 68 to get y ≈ 0.09(68) + 35.25, which gives us the predicted weight (or in our original context, diameter).
When dealing with correlation as in the case of the correlation coefficient of -0.56 between body weight and fuel efficiency, the coefficient of determination is found by squaring the correlation coefficient, which gives us 0.3136. This value tells us that approximately 31.36% of the variability in fuel efficiency can be explained by the linear relationship with body weight.