Does how long young children remain at the lunch table help predict how much they eat? Here are data on a random sample of $20$toddlers observed over several months. “Time” is the average number of minutes a child spent at the table when lunch was served. “Calories” is the average number of calories the child consumed during lunch, calculated from careful observation of what the child ate each day.

Here is some computer output from a least-squares regression analysis of these data. Do these data provide convincing evidence at the $\mathrm{\alpha }=0.01\mathrm{\alpha }=0.01$level of a linear relationship between time at the table and calories consumed in the population of toddlers?

$$\begin{gathered} \mathrm{Predictor}\mathrm{Coef}\mathrm{SE}\mathrm{Coef}\mathrm{T}\mathrm{P} \\ \mathrm{Constant}560.6529.3719.090.000 \\ \mathrm{Time}-3.07710.8498-3.620.002 \\ \mathrm{S}=23.3980\mathrm{R}-\mathrm{Sq}=42.1\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=38.9\% \end{gathered}$$

We need to

Consider:

$$\begin{gathered} \mathrm{n}=\mathrm{Sample}\mathrm{size}=20 \\ \mathrm{\alpha }=\mathrm{Significance}\mathrm{level}=0.01 \end{gathered}$$

The estimate of the slope ${\mathrm{b}}_1$is given in the row "Time" and in the column "Coef" of the given computer output:

${\mathrm{b}}_1=-3.0771$

The estimated standard deviation of the slope role="math" localid="1654164770191" ${\mathrm{SE}}_{\mathrm{b}1}$is given in the row "Time" and in the column "SE Coef" of the given computer output:

${\mathrm{SE}}_{\mathrm{b}1}=0.8498$

Given claim: Slope is nonzero:

The null hypothesis or the alternative hypothesis states the given claim The null hypothesis states that the slope is zero. If the given claim is the null hypothesis, then the alternative hypothesis states the opposite of the null hypothesis.

$$\begin{gathered} {\mathrm{H}}_0:{\mathrm{\beta }}_1=0 \\ {\mathrm{H}}_{\alpha }:{\mathrm{\beta }}_1\ne 0 \end{gathered}$$

Compute the value of the test statistic:

$\mathrm{t}=\frac{{\mathrm{b}}_1-{\mathrm{\beta }}_1}{{\mathrm{SE}}_{{\mathrm{b}}_1}}=\frac{-3.0771-0}{0.8498}\approx -3.6210$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the Student's T table in the appendix containing the -value in the row $\mathrm{df}=\mathrm{n}-2=20-2=18$We can ignore the minus sign in the test statistic:

$0.001=2(0.0005)<\mathrm{P}<2(0.001)=0.002$

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:
$\mathrm{P}<0.01\Rightarrow \mathrm{Reject}{\mathrm{H}}_0$