Lesson 11 Music to My Ears Practice Understanding

Learning Focus

Determine if the results of an experiment could have occurred by chance.

If the averages between two experimental groups are different, is that enough to declare the experiment a success?

Open Up the Math: Launch, Explore, Discuss

Mrs. Garcia is talking with her students who claim that if she allows them to listen to music on their test, they will perform better. Mrs. Garcia says that she is sure this is not true, but her students are insistent that it will work. Because they have been studying statistics in class, the students suggest that they conduct an experiment on the next test. Mrs. Garcia agrees that if they can design an experiment that might give convincing results, she will allow them to test their claim on the next test.

The students decide they will take the $30$ students in the class and split them in half. They will randomly assign $15$ students to take the exam without music, and the other half will take the exam while listening to music. The results of the two groups are given:

With Music
$78$	$83$	$95$	$62$	$81$
$84$	$80$	$83$	$77$	$76$
$92$	$72$	$85$	$88$	$86$

Without Music
$92$	$77$	$99$	$60$	$58$
$77$	$85$	$80$	$67$	$70$
$72$	$74$	$81$	$80$	$81$

1.

Create graphs and numerical summaries for each set of results for the experiment.

Music group graph:

Music group numerical summary:

No-music group graph:

No-music group numerical summary:

Does there appear to be evidence that listening to music helps students perform better on average on the test? Give evidence to support your answer.

2.

As the students are comparing the results, many students are convinced the results seem to back up their claim and think they should be allowed to use music on any exam. Mrs. Garcia is not so sure. She thinks they got lucky and by random chance, more of the students who would have scored higher anyway ended up in the music group. What do you think about Mrs. Garcia’s claim?

3.

One of the students suggests that they could use simulation to help them decide. They suggest that they take the $30$ scores, randomly split them up into two groups, and then find the average of the two groups and find the difference between the averages. They suggest they create a distribution by doing this over and over again, plotting the differences between the two groups, and seeing how often a difference as large or larger as the one they saw in the experiment happens. Use this data to carry out a simulation $6$ times. Based on your results, do you think that music helps students on the test? Justify your response.

4.

Combine your results with the rest of the class. When you look at the combined data for the rest of the class, how likely is a difference as large or larger than that observed in the experiment to happen?

5.

Suggest a way to answer this problem if you did not want to simulate taking samples over and over again. Use your method to determine if music helped the students on the test or not.

Ready for More?

How could you improve the experimental design to test whether listening to music during a test will improve test scores?

Takeaways

Simulations and experiments:

Lesson Summary

In this lesson, we used a simulation to determine if the results of an experiment could have occurred by chance. We found that simulation is useful not only for sampling but also for evaluating experimental results.

Retrieval

1.

Recall the three manipulations we can perform on the equations in a system.

Replace an equation in the system with a constant multiple of that equation.
Replace an equation in the system with the sum or difference of the two equations.
Replace an equation with the sum of that equation and a multiple of the other.

What was done to transform the first matrix into the second matrix?

$[\begin{array}{rrr} 6 & 6 & 45 \\ 3 & 2 & 19 \end{array}] \Rightarrow [\begin{array}{rrr} - 3 & - 3 & - 22.5 \\ 3 & 2 & 19 \end{array}]$

2.

Explain why when you type $y = \sin^{- 1} (- 0.5)$ into your calculator, it gives you a negative angle.