Chapter 12.1, Problem 19E

To determine

To find: that is there convincing evidence that listening to music at a louder volume hurts test performance.

Answer to Problem 19E

Yes

Explanation of Solution

Given:

  

  $\begin{array}{l}n=\text{30}\\ \alpha \text{=Significance level =0}\text{.05}\end{array}$

Claim: Slope is negative (hurts performance)

Formula used:

The test statistics

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

Calculation:

The calculation of the slope $b_1$ is mention in the row “Volume” and in the column “Coef” of the mention in the computer output:

  $b_1=-0.0483$

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

  $S{E}_{b_1}=0.0116$

Given claim: Slope is negative (hurts performance)

The null hypothesis or the alternative hypothesis statement is the given claim. The null hypothesis statement is 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.

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

The test statistic:

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}=\frac{-0.0483-0}{0.0116}=-4.1638$

The P-value is the probability of obtaining the value of the test statistic $df=n-2=30-2=28$ .

  $P<0.0005$

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

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

There is enough convincing proof that listening to music at a louder volume huts test performance.