Chapter 12.1, Problem 18E

To determine

To Explain: that there is convincing evidence of a negative linear relationship between researcher’ swim time and his pulse rate, yes or not.

Answer to Problem 18E

There is convincing proof that there is a negative linear relationship between researcher’s swim time and his pulse rate in the population of days on which he swims 2000 yards.

Explanation of Solution

Given:

  

  $\begin{array}{l}n=\text{Sample size =23}\\ \alpha \text{=Significance level =0}\text{.05}\end{array}$

Given claim: Slope is negative

Formula used:

The test statistics

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}$

Calculation:

Typing the Time data values in the first column of the spreadsheet of Minitab and type the Pulse data values in the second column.

Click on Start >Regression > Regression > Fit Regression model

Type the Pulse under Responses and type Time under Continuous predictors.

The resulting output is

The calculation of the slope $b_1$ is mentiion in the row Time and in the column coef of the mention computer output:

  $b_1=-9.69$

The calculated standard deviation of the slope $S{E}_{b_1}$ is mention in the row Time and in the column Coef of the mention computer output:

  $b_1=-9.69$

The calculated standard deviation of the slope $S{E}_{b_1}$ is mention in the row Time and in the column SE Coef of the mention computer output:

  $S{E}_{b_1}=1.89$

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

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1<1\end{array}$

The test statistic:

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}=\frac{-9.69-0}{1.89}=-5.1270$

The P-value is the probability of getting the value of the test statistic. $df=n-2=23-2=21$

  $P<0.0005$

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

  $P<0.05\Rightarrow \text{Reject }{H}_0$

There is convincing proof that there is a negative linear relationship between researcher’s swim time and his pulse rate in the population of days on which he swims 2000 yards.