Question

Music and mazes A researcher wishes to determine if people are able to complete a certain pencil and paper maze more quickly while listening to classical music. Suppose previous research has established that the mean time needed for people to complete a certain maze (without music) is 40 seconds. The researcher decides to test the hypotheses $H_0:\mu =40$ versus $H_a:\mu <40$ where $\mu =$ the time in seconds to complete the maze while listening to classical music. To do so, the researcher has $10,000$ people complete the maze with classical music playing. The mean time for these people is $\overline{x}=39.92$ seconds, and the P-value of his significance test is $0.0002$ Explain why this result is statistically significant, but not practically important.

Short answer

Because the $P-$value is low, the result is static. Because a $0.08-$second improvement is minor, the outcome is unimportant in practice.

Step-by-step solution

Given information

$$\begin{gathered} H_0:\mu =40 \\ {H}_1:\mu <40 \\ \overline{x}=39.92 \\ P-value=0.0002 \end{gathered}$$

Calculation

Assume that the level of significance $\alpha$is $0.05$

$\alpha =0.05$

Reject the null hypothesis $H_0$if the$P-$value is less than the significance level, and the result is statistically significant.

$0.0002<0.05\Rightarrow Statisticallysignificant$

However, the finding is not practically significant because the improvement was just $40-39.92=0.08$ seconds, and $0.08$ seconds is not a significant improvement in the maze completion time.