this situation. Calculate a significance probability. Do these results warrant rejecting the null hypothesis at a significance level of or = 0.05? 3. A researcher wanted to see if live reggae music improved students' math test scores. ' He selected a sample of 61 students from his university and gave them a math test. The students then studied for two and a half hours while an acoustic reggae band played quietly. The students then took another math test of the same difficulty. The researcher's variable was the change in test score for each student a positive change meant the student did better, while a negative change meant a student did worse. He found that the changes had an approximately normal distribution with mean 6.5 and standard deviation 12 . (a) (5 points ) Assuming the sample is random , test the hypothesis that the population average is positive. (b) (2 points) Does the study provide convincing evidence that live reggae music improves students' math test scores? Explain why or why not. 4. Every semester, a professor gives his undergraduate statistics classes a test for psychic powers. The test consists s of guessing which side of a screen a picture will appear on: left or right . In one trial of the test , a student has to guess "Left " or "Right "; then R's random number generator will randomly choose one side of the screen to display a picture of a star . T student repeats the process for a total of 20 trials . (a) (1 point) In words, the null hypothesis for a particular student is that they don't have psychic powers and they're just randomly guessing. The alternative hypothesis is that they do have psychic powers and in the long run, they can do better than randomly guessing Let p be the probability the student guess correctly on any particular trial . Write down mathematical null and alternative hypotheses in terms of p. (b) (1 point ) Suppose the null hypothesis is true for a particular student . Let Y be the number of times the student guesses correctly. Then if the null is true, Y has a Binomial (20 , 0.5) distribution Which of the following is true? i. If the student guesses 13 right out of 20, the significance probability ( P-value) is the probability under the null of guessing 13 or more out of 20. ii. If the student guesses 13 right out of 20, the significance probability (P-value) is the probability under the null of guessing 13 or fewer out of 20. Hint: Remember that the smaller the P-value, the stronger the evidence for the alter - native hypothesis (c) (3 points ) Suppose a student guesses 13 right out of 20. What is the P-value? Do you think 13 out of 20 is intriguing evidence that the student has psychic powers? (Use Binomial probability)