Question

Time at the table Refer to Exercise 14.

(a) Construct and interpret a 98% confidence interval for the slope of the population regression line. Explain how your results are consistent with the significance test in Exercise 14.

(b) Interpret each of the following in context:

(i)s

(ii) $r^2$

(iii) The standard error of the slope

Short answer

a) $(-5.2458,-0.9084)$

(b)

(i) $s=23.3980$indicates that the expected error when making predictions using the regression line is about $23.3980$calories.

(ii) $r^2=42.1\%$indicates that 42.1% of the variation between the variables has been explained by the linear regression line.

(iii) $S{E}_b=0.8498$indicates that the estimation of the slope is expected to vary on average by 0.8498 over all possible samples.

Step-by-step solution

Part (a) Step 1: Given Information 

Given:

$$\begin{gathered} n=20 \\ b=-3.0771 \\ S{E}_b=0.8498 \end{gathered}$$

Part (a) Step 2: Explanation

The degrees of freedom is the sample size decreased by 2 :

$$\begin{gathered} df=n-2 \\ =20-2 \\ =18 \end{gathered}$$

The critical t-value can be found in table B in the row of d f=18 and in the column of

$t^{*}=2.552$

The boundaries of the confidence interval then become:

localid="1650694848183" $$\begin{gathered} b-t^{*}\times S{E}_b \\ =-3.0771-2.552\times 0.8498 \\ =-5.2458 \\ b+t^{*}\times S{E}_b \\ =-3.0771+2.552\times 0.8498 \\ =-0.9084 \end{gathered}$$

Part (b) Step 1: Given Information

Given:

$$\begin{gathered} s=23.3980 \\ {r}^2=42.1\% \\ =0.421 \\ S{E}_b=0.8498 \end{gathered}$$

Part (b) Step 2: Explanation

(i) The population standard deviation is commonly denoted by the letter $s$(the Greek lower-case letter "sigma"). The standard deviation of a sample of scores is denoted by the letter $s$.

(ii) $r^2$is a statistical measure that quantifies the proportion of variation explained by an independent variable or variables in a regression model for a dependent variable.

(iii) The standard error of the computed value of b is$S{E}_b$By dividing the value of $b$by its standard error, a $t$-test for statistical significance of the coefficient is performed.