Question

Sorting the music Student researchers Adam, Edward, and Kian wondered

if music would affect performance for certain tasks. To find out, they had student volunteers sort a shuffled set of 26 playing cards by face value and by color. Nineteen of the 38 volunteers were randomly assigned to listen to music during the sorting, while the others listened to no music. Here are parallel boxplots of the time in seconds that it took to sort the cards for the students in each group:

Do these data give convincing evidence of a difference in the true mean sorting times at the α=0.10$\frac{305}{1526}=0.200=20.0\%\alpha =0.10$significance level?

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameters of interest.

b. Check if the conditions for performing the test are met.

Short answer

Part a) The hypothesis is

$$\begin{gathered} H_0:{\mu }_1={\mu }_2 \\ {H}_a:{\mu }_{1}notequalto{\mu }_2 \end{gathered}$$

Part b) All conditions are not met.

Step-by-step solution

Part a) Step 1: Explanation

The given claim is that a difference in the means.

Now we must determine the most appropriate hypotheses for a significance test.

As a result, either the null hypothesis or the alternative hypothesis is the claim. According to the null hypothesis, the population proportions are equal. If the claim is the null hypothesis, the alternative hypothesis is the polar opposite of the null hypothesis.

Therefore, the appropriate hypotheses for this are:

$$\begin{gathered} H_0:{\mu }_1={\mu }_2 \\ {H}_a:{\mu }_{1}notequalto{\mu }_2 \end{gathered}$$

Where we have,

${\mu }_1=$the true average sorting time of music listeners.

${\mu }_2=$ the true mean sorting time of non-music listeners.

Part b) Step 1: Explanation

There are three requirements that must be met:

Because the volunteers were assigned to treatment at random, it is satisfactory.

Independent: It is satisfying because the $19$people make up less than $10\%$of the total population.

Normal: It is not satisfactory because the sample size for no music is small, and the distribution of No music is not approximately Normal due to a strong rightward skewed distribution.

As a result, all of the conditions are not met because the large sample condition is not met, and a hypothesis test for the mean difference is not appropriate.