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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 X .
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.
b.
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.
Z-Test for Sample Means
c.
z
X 

 11.62
N
d.
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.
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.  1.96 
524  
 528.05    519.05
100
2345
2.
a.
b.
c.
d.
H1 = X   ; H 0 = X  
One-Sample T-Test
df = 6; Critical t = -1.943
t = -2.49, which is less than our critical t and in our rejection region, therefore we would reject H 0
and conclude that right hemisphere brain damage does significantly reduce spatial skills.
e.
 2.447 
a.
b.
c.
d.
H1 = μD ≠ 0; H1 = μD = 0
Repeated-Measures T-Test
df = 6; Critical t = ±2.447
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.
11.71  
 14.92    8.50
3.4756
7
3.
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.