6.2 Simple linear regression

Figure 6.1 shows a scatter plot of birth weight (grams) against gestational age (in weeks) at birth for 24 newborn babies. Gestational age shows time since conception. The straight line shows the linear relationship between the two variables.

1. 1.

   Is there evidence of a positive relationship between birth weight and gestational age? If so, what is this relationship?

2. 2.

   Can we predict the birth weight for a child born at 34 weeks? What is a 95% confidence interval for this prediction?

The answers to these questions relate to the intercept and slope of the line on Figure 6.1, as well as the random variation in birth weights for a given gestation period. Why might such random variation occur?

Other unobserved variables will have an affect on birth weights, e.g. genetics, mother’s diet, whether or not the mother smokes,…

Does outside temperature affect household gas consumption? In the 1960’s the weekly gas consumption of a single household was recorded over the period of one year. Figure 6.2 shows a scatter plot of weekly gas consumption (1000’s cubic feet) against the average weekly outside temperature (${}^{\circ }$C).

1. 1.

   Is a linear relationship reasonable?

2. 2.

   Is there a significant decrease in gas consumption for every $1^{\circ }$C increase in average weekly temperature?

Can a mammal’s body weight be used to predict its brain weight? Figure 6.3 shows a scatter plot of brain weight (g) against body weight (kg) for a sample of 62 species of mammals, and a plot of log(brain weight) against log(body weight).

1. 1.

   Which measurement scale is more appropriate for a linear relationship?

2. 2.

   Can we predict the brain size of a zebra weighing 290kg (zebras were not included in the sample)?

3. 3.

   Can we assess how well the linear model fits the data?