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Question 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.

Part A

Describe the relationship between height and weight.

The relationship is linear and is positive and reasonably strong.

Part B

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

The equation is as follows:

\(weight = -105.0113 + 1.0176 * height\)

The model tells us that for every centimeter in height, we expect an additional 1.02 kg in weight. The intercept doesn’t have any intrinsic meaning here because if height == 0cm, the model would give us a weight of -105 kg. As such, it serves only as an adjustment.

Part 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.

\(H0: True\: slope\: height\: is\: zero\: (\beta1 = 0)\)

\(HA: True\: slope\: greater\: than\: zero\: (\beta1 > 0)\)

The p-value for this test is incredibly small as shown in the table above. We can reject the null and say that the true slope parameter is >0.

Part D

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

r <- 0.72
r.square <- r^2
r.square
## [1] 0.5184

The r.square is 0.5184 and tells us that 51.84% of the variability around the mean in weight is explained by height.