Nicole and Elena wanted to know if listening to music at a louder volume negatively impacts test performance. To investigate, they recruited $30$volunteers and randomly assigned $10$volunteers to listen to music at $30$decibels, $10$volunteers to listen to music at $60$decibels, and $10$volunteers to listen to music at $90$decibels. While listening to the music, each student took a 10-question math test. Here is computer output from a least-squares regression analysis using role="math" localid="1654167255833" $\mathrm{x}=\mathrm{volume}$and $\mathrm{y}=\mathrm{number}$correct:

$$\begin{gathered} \mathrm{Predictor}\mathrm{Coef}\mathrm{SE}\mathrm{Coef}\mathrm{T}\mathrm{P} \\ \mathrm{Constant}9.90000.752513.1560.0000 \\ \mathrm{Volume}-0.04830.0116-4.1630.0003 \\ \mathrm{S}=1.55781\mathrm{R}-\mathrm{Sq}=38.2\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=36.0\% \end{gathered}$$

Is there convincing evidence that listening to music at a louder volume hurts test performance? Assume the conditions for inference are met.

We need to find convincing evidence that listening to music at a louder volume hurts test performance.

Consider:

$$\begin{gathered} \mathrm{n}=\mathrm{Sample}\mathrm{size}=30 \\ \mathrm{\alpha }=\mathrm{Significance}\mathrm{level}=0.05 \end{gathered}$$

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

${\mathrm{b}}_1=-0.0483$

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

${\mathrm{SE}}_{\mathrm{b}1}=0.0116$

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:

$\mathrm{t}=\frac{{\mathrm{b}}_1-{\mathrm{\beta }}_1}{{\mathrm{SE}}_{{\mathrm{b}}_1}}=\frac{-0.483-0}{0.0116}\approx -4.1638$

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 $\mathrm{df}=\mathrm{n}-2=30-2=28$We can ignore the minus sign in the test statistic:

$\mathrm{P}<0.005$

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:
$\mathrm{P}<0.05\Rightarrow \mathrm{Reject}{\mathrm{H}}_0$