7.35 Body measurements, Part IV.

The scatterplot and least squares summary below show the relationship between weight measured in kilograms and height measured in centimeters of 507 physically active individuals.

(a)Describe the relationship between height and weight.

Ans: Looking at the catterplot ,looks like it is positive corelation, moderate to-strong corelation and linear relationship. There are a few outliers but no points that appear to be influential.

(b)Write the equation of the regression line. Interpret the slope and intercept in context.

Ans:

First column gives model parameters \(\beta0\) and \(\beta1\)

\(\hat{weight}\) = -105.0113 + 1.0176 * height

Slope: For each additional centimeter in height,the model predicts the average weight to be 1.0176 additional kilograms (about 2.2 pounds).

Intercept: People who are 0 centimeters tall are expected to weigh -105.0113 kilograms. This is obviously not possible. Here, the y-intercept serves only to adjust the height of the line and is meaningless by itself.

(c)Do the data provide strong evidence that an increase in height is associated with an increase in weight? State the null and alternative hypotheses, report the p-value, and state your conclusion.

Ans: \(H_0\): The true slope coefficient of height is zero (\(\beta1\) = 0).

\(H_A\): The true slope coefficient of height is greater than zero (\(\beta1\) > 0).

A two-sided test would also be acceptable for this application. The p-value for the two-sided alternative hypothesisis (\(\beta1\) \(\neq\) 0) is incredibly small, so the p-value for the onesided hypothesis will be even smaller. That’s why , we reject \(H_0\). The data provide convincing evidence that height and weight are positively correlated. The true slope parameter is indeed greater than 0.

(d)The correlation coefficient for height and weight is 0.72. Calculate R2 and interpret it in context.

Ans: \(R^2\) = \(0.72^2\) = 0.52. Approximately 52% of the variability in weight can be explained by the height of individuals.