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.
To answer this question, we will go through the five steps of hypothesis testing:
Step 1: State the hypotheses.
The null hypothesis (H0) states that there is no significant difference in brain activity between listening to music while solving math problems and not listening to music while solving math problems. The alternative hypothesis (Ha) states that there is a significant difference, specifically an increase in brain activity when listening to music while solving math problems.
Step 2: Formulate an analysis plan.
We will use a one-sample z-test because we know the population mean and standard deviation, and we have a single sample to compare it with.
Step 3: Analyze sample data.
The sample data is given as a percentage signal change of 58 in the brain area of interest.
Step 4: Interpret the results.
To interpret the results, we will calculate the z-score of the sample data and compare it with the critical value for a significance level of .01.
Calculating the z-score:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
z = (58 - 35) / (10 / sqrt(1))
z = 23 / 10
z = 2.3
Step 5: Make a decision.
We will compare the calculated z-score of 2.3 with the critical value for a significance level of .01.
Using a standard normal distribution table or a statistical software, the critical value for a .01 significance level (two-tailed test) is approximately ±2.576.
The z-score of 2.3 is less than 2.576, which means it does not fall in the critical region. Therefore, we fail to reject the null hypothesis.
In simple terms, the researcher should conclude that there is not enough evidence to support the hypothesis that listening to music while solving math problems significantly increases brain activity in the particular area of interest.
Please note that the sketch showing the comparison distribution, cutoff(s), and the score of the sample on this distribution would require a visual representation.
Ho: change 1 = change 2
Ha: change 1 ≠ change 2
Z = (score-mean)/SD = (58-35)/10 = ?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion in smaller area = Z score.
The rest is up to you.