A Researcher Predicts That Listening To Music While Solving
18 A Researcher Predicts That Listening To Music While Solving Math
A researcher predicts that listening to music while solving math problems will make a particular brain area more active. To test this, a research participant has her brain scanned while listening to music and solving math problems, and the brain area of interest has a percentage signal change of 58. From many previous studies with this same math problems procedure (but not listening to music), it is known that the signal change in this brain area is normally distributed with a mean of 35 and a standard deviation of 10. (a) Using the .01 level, what should the researcher conclude? Solve this problem explicitly using all five steps of hypothesis testing, and illustrate your answer with a sketch showing the comparison distribution, the cutoff (or cutoffs), and the score of the sample on this distribution. (b) Then explain your answer to someone who has never had a course in statistics (but who is familiar with mean, standard deviation, and Z scores).
Paper For Above instruction
Introduction
The question posed by the researcher is whether listening to music while solving math problems results in increased brain activity in a specific brain region. This is a classic hypothesis testing scenario where an observed effect (signal change of 58) is compared against a known population distribution under the assumption of no effect. The goal is to determine if the data provides sufficient evidence to support the alternative hypothesis that music increases brain activity, using a significance level of 0.01.
Step 1: State the hypotheses
The null hypothesis (H0) assumes that listening to music does not affect brain activity, meaning the mean signal change remains at the known population mean of 35. The alternative hypothesis (H1) suggests that music increases brain activity, implying that the mean signal change with music is greater than 35.
\[
H_0: \mu = 35
\]
\[
H_1: \mu > 35
\]
This is a one-tailed test because the researcher predicts an increase.
Step 2: Set the significance level
The significance level, \(\alpha\), is 0.01, indicating that the researcher is willing to accept a 1% chance of rejecting the null hypothesis when it is actually true (Type I error).
Step 3: Calculate the test statistic
The test statistic for a mean against a known population mean with a known standard deviation is a Z-score:
\[
Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}
\]
Since the problem states the brain signal change is normally distributed with known parameters, and the observed value is 58, the calculation simplifies to:
\[
Z = \frac{58 - 35}{10 / \sqrt{n}}
\]
However, the sample size \(n\) is not specified in the problem; the typical assumption is \(n=1\) (since a single participant's brain scan is used). With \(n=1\):
\[
Z = \frac{58 - 35}{10} = \frac{23}{10} = 2.3
\]
If the sample size was larger, the denominator would change accordingly, but under the typical assumption of a single observation, \(Z=2.3\).
Step 4: Determine the cutoff value
For a one-tailed \(\alpha=0.01\) test, find the critical Z-value:
\[
Z_{critical} \approx 2.33
\]
This means that if the calculated Z exceeds 2.33, the result is statistically significant at the 1% level.
Step 5: Make a decision
Since the computed Z-value is 2.3, which is slightly less than 2.33, it does not quite reach the critical value. Therefore, we fail to reject the null hypothesis at the 0.01 significance level.
Conclusion:
Based on this analysis, there is insufficient evidence at the 1% significance level to conclude that listening to music causes an increase in brain activity in this region. The observed signal change (58) is not statistically significantly higher than what would be expected under the null hypothesis.
Sketch Explanation
Imagine a normal distribution curve centered at 35 (the null mean). The critical cutoff point on this distribution corresponds to a Z of 2.33, which is approximately at 58 (since \(\text{cutoff} = \mu + Z_{critical} \times \sigma / \sqrt{n}\)), illustrating the threshold for significance. Our observed value of 58 aligns with this cutoff, but because the calculated Z is just below 2.33, it falls just short of the rejection region.
Explanation for a Non-Statistic Audience
Think of the brain activity level as a score on a test that usually averages around 35. We want to see if listening to music makes this score higher. To do this, we compare the score of 58 to what we'd expect if music had no effect—that is, if the score just randomly fluctuates around 35. When we calculate how unusual a score of 58 is, based on the usual variation, we find that it’s somewhat higher but not quite high enough to confidently say "Yes, music really boosts brain activity." Because our criteria are very strict (we only accept a 1% chance of being wrong), this result isn't strong enough to claim a real effect. Essentially, the observed high score could still just be due to normal fluctuations, so we conclude there’s not enough evidence to say music definitely increases brain activity in this case.
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