a research participant has her brain scanned while listening to music and solving math problems, and the brain area of interest has a percent 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. 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. 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).
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Ho: u = 35
H1: u > 35
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test stat: z(58) = (58-35)/10 = 23/10 = 2.3
58 is 2.3 standard deviations above the hypothesized mean of 35
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Test results this far or further above the mean have slightly more
than 1% chance of occuring.
You could claim, with nearly 99% assurance, that the music helped.
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Cheers,
Stan H.
Good job.
To solve this problem, we need to follow the five steps of hypothesis testing:
Step 1: State the hypotheses.
In this case, the null hypothesis states that the mean signal change in the brain area of interest is equal to 35 (µ = 35). The alternative hypothesis states that the mean signal change is greater than 35 (µ > 35).
Step 2: Formulate an analysis plan.
We need to calculate the test statistic and compare it to a critical value.
Step 3: Analyze sample data.
Calculate the test statistic using the formula: z = (x - µ) / σ, where x is the sample mean (58), µ is the hypothesized mean (35), and σ is the standard deviation (10).
Using these values, we can calculate z as (58 - 35) / 10, which equals 23 / 10 = 2.3.
Step 4: Interpret the results.
We need to compare the test statistic to the critical value to determine if we should reject the null hypothesis. Since the test is one-tailed and we are interested in values greater than 35, we will use the .01 level to determine our critical value. The critical value for a one-tailed test with a .01 level of significance is approximately 2.33.
Now we can compare the test statistic (2.3) to the critical value (2.33). Since the test statistic falls below the critical value, we do not have enough evidence to reject the null hypothesis.
Step 5: Draw a conclusion.
Based on our analysis, we conclude that there is insufficient evidence to claim that listening to music significantly increases the signal change in the brain area of interest.
To illustrate this, imagine a bell-shaped curve called the normal distribution. The curve represents the distribution of signal changes in the brain area without listening to music (µ = 35, σ = 10). The critical value of 2.33 lies on the right tail of the curve. The test statistic of 2.3 lies slightly before the critical value, indicating that it is within the range of what we would expect to see if the null hypothesis were true. This means that the observed signal change of 58 could reasonably occur due to random chance alone. Therefore, we do not have enough evidence to support the claim that listening to music significantly increases the signal change in this brain area.
In simple terms, the research participant's brain scan showed a signal change of 58. By comparing it to what we would expect based on previous studies (µ = 35, σ = 10), we determined that the signal change is 2.3 standard deviations above the hypothesized mean. However, this value is not high enough to be considered significant and is within the range of what we could expect by chance. Therefore, we cannot conclude that listening to music has a significant effect on the signal change in this brain area.