Saturday, March 23, 2013

Individual Assignment


11. List the five steps of hypothesis testing, and explain the procedure and logic of each.
Step 1: Aron, Aron, and Coups, (2009) “restate the question as a research hypothesis and a null hypothesis about the populations” (p. 115). Here one makes a research hypothesis concerning a predicted relation among populations. The null and research hypothesizes are the opposite of each other. A correct research hypothesis means the null hypothesis cannot be correct, and a correct null hypothesis means the research hypothesis cannot be correct.
Step 2: Aron, Aron, and Coups, (2009) “determine the characteristics of the comparison distribution” (p. 115). By reaching a true null hypothesis the population situation represents the comparison distribution, which is the distribution compared to the score and based on the results of the sample.
Step 3: Aron, Aron, and Coups, (2009) “determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected” (p. 115). Here one rejects the null hypothesis if the point of the cutoff sample score reaches or exceeds the sample score. If the null hypothesis is true the Z score is set at a score, which would be unlikely.
Step 4: Aron, Aron, and Coups, (2009) “determine your sample’s score on the comparison distribution” (p. 115). Here one gathers the test’s sample results.
Step 5: Aron, Aron, and Coups, (2009) “decide whether to reject the null hypothesis” (p. 115). Here one either declares the test invalid or rejects the null hypothesis by comparing the cut off Z score to the sample’s Z score.
14. Based on the information given for each of the following studies, decide whether to reject the null hypothesis. For each, give (a) the Z-score cutoff (or cutoffs) on the comparison distribution at which the null hypothesis should be rejected, (b) the Z score on the comparison distribution for the sample score, and (c) your conclusion. Assume that all populations are normally distributed.
Population
Study              µ          σ         Sample score           p                Tails of Test
A                                 5          1                 7                   .05                1 (high predicted)
B                                 5          1                 7                   .05                2
C                                 5          1                 7                   .01                1 (high predicted)
D                                 5          1                 7                   .01                2
A) (a) 1.645 Z score cutoff, (b) Z = 2 (c) reject the null hypothesis
B) (a) 1.96 Z score cutoff,  (b) Z = 2, (c) reject the null hypothesis,
C) (a) 2.3263 Z score cutoff, (b) Z = 2, (c) fail to reject the null hypothesis,
D) (a) 2.576 Z score cutoff, (b) Z = 2 (c) fail to reject the null hypothesis,
18. 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).
Population
Study             µ                      σ          Sample Score             p                      Tails of Test
A                     100.0               10.0                 80                    .05                   1 (low predicted)
B                     100.0               20.0                 80                    .01                   2
C                     74.3                 11.8                 80                    .01                   2
D                     16.9                 1.2                   80                    .05                   1 (low predicted)
E                      88.1                 12.7                 80                    .05                   2
(A)  The researcher should conclude whether there is or is not sufficient statistical evidence that music increases the math problem solving skills.  
Step 1: Aron, Aron, and Coups, (2009) “restate the question as a research hypothesis and a null hypothesis about the populations” (p. 115).
Does listening to music while solving math problems make a particular brain area more active?
Population 1: Music increases the math problem solving skills.
Population 2: Music has no effect on math problem solving skills.
Step 2: Aron, Aron, and Coups, (2009) “determine the characteristics of the comparison distribution” (p. 115).
It is assumed that music increases the math problem solving skills. The null hypothesis is “Music has no effect on math problem solving skills.” The comparison distribution is population two’s distribution.  
Step 3: Aron, Aron, and Coups, (2009) “determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected” (p. 115).
Reject the null hypothesis if the music has no effect on math problem solving skills score is within the bottom or the top 2.5% of the comparison distribution. The cutoff Z scores for the 1% level are -2.33 or 2.33.     
Step 4: Aron, Aron, and Coups, (2009) “determine your sample’s score on the comparison distribution” (p. 115).
Z = (x - m)/s = (58 - 35)/10 = 2.30      
Step 5: Aron, Aron, and Coups, (2009) “decide whether to reject the null hypothesis” (p. 115).
Since 2.30 < 2.3263, therefore fails to reject the null hypothesis. 
(b) Seeing that the p-value is greater than 1%, one fails to reject the null hypothesis. The result of this one test is not sufficient evidence to reject the belief that the mean percentage is 0.35.  



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