a) To construct a 95% confidence interval for β 1 the slope of the regression line, we use the following formula: b_1 \pm t_{\alpha/2, n-2} \cdot s_b.
b) The confidence interval for β1 tells us that we can be 95% confident that the true slope of the regression line is between 5.203 and 9.181.
To construct a 95% confidence interval for β 1 , the slope of the regression line, we use the following formula:
b_1 \pm t_{\alpha/2, n-2} \cdot s_b
where:
b 1 is the estimated slope coefficient from the regression output
t α/2,n−2 is the t-value for the desired level of confidence (α=0.05 for a 95% confidence interval) and degrees of freedom (n−2, where n is the sample size)
s is the standard error of the slope coefficient
From the regression output, we can see that b 1=7.1919 and s
b =0.9531. The degrees of freedom are n−2=8−2=6.
Using the t.dist() function in R, we can find the t-value for a 95% confidence interval with 6 degrees of freedom:
This gives us a t-value of 2.447.
Now we can plug all of these values into the formula to construct the confidence interval:
7.1919 \pm 2.447 \cdot 0.9531
This gives us the following 95% confidence interval for β 1 :(5.203, 9.181)
(b) The confidence interval for β1 tells us that we can be 95% confident that the true slope of the regression line is between 5.203 and 9.181.
In other words, we can be 95% confident that for every 1 unit increase in snake length, there is an increase in weight between 5.203 and 9.181 units.
In the context of this setting, this means that we can be 95% confident that longer snakes tend to weigh more.
This is biologically plausible, as larger snakes need more energy to maintain their larger bodies.
It is important to note that the confidence interval is only valid for the population of snakes from which the sample was drawn.