Homework #3



Homework #3

Please complete all of the problems to obtain credit. The answers are provided on the last page, to obtain credit, you must show that you’ve done more than simply to copy the answers from the last page.

1. In 1979 a group of students in Arizona took an experimental class that was designed to improve grades on the math portion of the SAT. These 2345 students from Arizona who subsequently took the SAT had a mean of 524 (on the math portion). Are these scores significantly higher than a population mean of 500 if we assume that σ = 100? Use a one-tailed test with α = .05 to evaluate your answers to the following questions?

a. State both in words, and then symbolically what your H1 and H0 would be.

b. What kind of test would you use (i.e. Repeated-Measures t-test, z-test, etc.)?

c. What is the probability of obtaining the mean that was obtained in this sample?

d. Based on the above information, would you reject H0, or fail to reject it? Why? How would you state your conclusion in words?

e. Compute the 95% confidence limits on μ.

2. On a standardized spatial skills task, normative data reveal that people typically get 15 correct solutions. A psychologist tests 7 individuals who have brain injuries in the right cerebral hemisphere. For the following data, determine whether or not right-hemisphere damage results in significantly reduced performance on the spatial skills task. Test with α set at .05 with one tail. The data are as follows: 12, 16, 9, 8, 10, 17, 10.

a. State symbolically what your H1 and H0 would be.

b. What kind of test would you use (i.e. Repeated-Measures t-test, z-test, etc.)?

c. What is your df? What is your critical t?

d. Based on the above information, would you reject H0, or fail to reject it? Why? What would your conclusion be?

e. Compute the 95% confidence limits on [pic].

3. Sensory isolation chambers are used to examine the effects of mild sensory deprivation. The chamber is a dark, silent tank where subjects float on heavily salted water and are thereby deprived of nearly all external stimulation. Sensory deprivation produces deep relaxation and had been shown to produce temporary increases in sensitivity for vision, hearing, touch, and even taste. The following data represent hearing threshold scores for a group of subjects who were tested before and immediately after one hour of deprivation. A lower score indicates more sensitive hearing. Do these data indicate that deprivation has a significant effect on hearing threshold? Test at the .05 level of significance with two tails.

|Subject |Before |After |

|A |31 |30 |

|B |34 |31 |

|C |29 |29 |

|D |33 |29 |

|E |35 |32 |

|F |32 |34 |

|G |35 |28 |

a. State symbolically what your H1 and H0 would be.

b. What kind of test would you use (i.e. Repeated-Measures t-test, z-test, etc.)?

c. What is your df? What is your critical t?

d. Based on the above information, would you reject H0, or fail to reject it? Why? What would your conclusion be?

4. Assuming in Question #1 that we used a smaller sample size, what effect would this probably have on our p-value? What about whether we reject or fail to reject the null hypothesis?

5. Given the First Commandment of Statistics (see the notes for the lecture on Hypothesis Testing), what can we conclude from the results of our experiment in Question #2 and what can we definitely not conclude?

6. In Question #1, if we were to obtain a probability of obtaining our sample equal to 6%, would we still reject or fail to reject the null hypothesis? Would it influence our conclusions at all? If so, what else would we say about this result?

7. Assume that in Question #3 that we used 5 different means of measuring hearing threshold (i.e. tones of a certain volume, calling the subjects name, and a loud crash, etc.), and evaluated our hypothesis with 10 different statistical tests. How would this effect your alpha? Using this new value of alpha, what would be your new critical value(s)?

Answers

1.

a. H1 = Our sample of students from Arizona will outperform the average student taking the SAT; x > μ.

H0 = Our sample of students from Arizona will perform at the same level or less than the average student taking the SAT; x ≤ μ.

b. Z-Test for Sample Means

c. [pic]

Table E10 only goes up to a z-score of 4, so we can assume that p < .0000. See what effect large sample sizes has on our p-value.

d. Reject H0, because the probability of obtaining our sample mean is less than 5%

and falls within our rejection region, which is above μ (because we are using a one-tailed test.

e. [pic]

2.

a. H1 = [pic]; H0 = [pic]

b. One-Sample T-Test

c. df = 6; Critical t = -1.943

d. t = -2.49, which is less than our critical t and in our rejection region, therefore we would reject H0 and conclude that right hemisphere brain damage does significantly reduce spatial skills.

e. [pic]

3.

a. H1 = μD ≠ 0; H1 = μD = 0

b. Repeated-Measures T-Test

c. df = 6; Critical t = ±2.447

d. t = 2.06, which is neither above 2.447, nor below -2.447, therefore we would fail to reject H0 and conclude that the data show no significant change in hearing threshold.

4. It would increase and would make it less likely for us to reject the null hypothesis.

5. We can conclude that it is highly unlikely that the results of our experiment are due to sampling error and very likely that with another sample that we’d get a similar result. We cannot conclude that our sample is very different from the population or anything about the magnitude of the difference between our sample and the population.

6. We would still reject the null hypothesis, but would add that our results indicate that some effect may be present, but that we may not be able to detect it due to chance.

7. We would decrease our alpha by dividing it by 5x10 = 50. Our new alpha, after Bonferroni Correction would be .001, and our new critical t, with df = 6, would be 5.959.

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