8.31 Body measurements, Part IV: The scatterplot and least squares summary below show the relation-ship between weight measured in kilograms and height measured in centimeters of 507 physically active individuals.
Data606 <- data.frame(
heading <- c("Intercept","height"),
Estimate <- c("-105.0113","1.0176"),
Std.Error <- c("7.5394","0.0440"),
Tvalue <- c("-13.93","23.13"),
Pr <- c("0.000","0.000")
)
colnames(Data606) <- c("heading","Estimate","Std.Error","T value","Pr")
knitr::kable(Data606)
| Intercept |
-105.0113 |
7.5394 |
-13.93 |
0.000 |
| height |
1.0176 |
0.0440 |
23.13 |
0.000 |
(a) Describe the relationship between height and weight.
Ans)
Predictor variable: height.
Response variable: weight.
The relationship is a positive strong correlation between height and weight.
(b) Write the equation of the regression line. Interpret the slope and intercept in context.
Ans)
weight^=−105.0113+1.0176∗height
slope= 1.0176
This indicates that each additional cm of height is associated with 1.0176 kg of weight.
intercept = -105.0113
This indicates that if the linear model is correct, then a height of 0 cm is associated with a weight of -105 kg which doesn’t make sense.
(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(o): The true slope coefficient of height is zero.
H(A): The slope coefficient of height is differe than zero.
Since the p-value is incredibly small we can reject the null hypthesis.The data has significant evidence that height and weight are correlated.
(d) The correlation coefficient for height and weight is 0.72. Calculate R2 and interpret it in context.
Ans)
R2=(0.72)^2=0.52
This means that 52% of the variability in weight can be explained by the height variable.