Question

Here are data on the time (in minutes) Professor Moore takes to swim $2000$yards and his pulse rate (beats per minute) after swimming on a random sample of $23$days:

Is there convincing evidence of a negative linear relationship between Professor Moore’s swim time and his pulse rate in the population of days on which he swims $2000$yards?

Short answer

Yes, convincing evidence of a negative linear relationship between Professor Moore’s swim time and his pulse rate in the population of days on which he swims $2000$yards.

Step-by-step solution

Given Information

We need to find convincing evidence of a negative linear relationship between Professor Moore’s swim time and his pulse rate in the population of days on which he swims $2000$yards.

Simplify

Consider:

$$\begin{gathered} \mathrm{n}=\mathrm{Sample}\mathrm{size}=23 \\ \mathrm{\alpha }=\mathrm{Significance}\mathrm{level}=0.05 \end{gathered}$$

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

${\mathrm{b}}_1=-9.69$

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

${\mathrm{SE}}_{\mathrm{b}1}=1.89$

Coefficients

Given claim: Slope is negative (reduction):

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:

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

Compute the value of the test statistic:

$\mathrm{t}=\frac{{\mathrm{b}}_1-{\mathrm{\beta }}_1}{{\mathrm{SE}}_{{\mathrm{b}}_1}}=\frac{-9.69-0}{1.89}\approx -5.1270$

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 $\mathrm{df}=\mathrm{n}-2=23-2=21$We can ignore the minus sign in the test statistic:

$\mathrm{P}<0.0005$

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

$\mathrm{P}<0.0005\Rightarrow \mathrm{Reject}{\mathrm{H}}_0$