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Psychology 406
Assignment \#6
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1) Hays --- Chapter 6, \#22; Chapter 7, \#26
\bigskip
2) Suppose you develop a test of spatial reasoning and administer it
to all second grade children in the Champaign/Urbana public schools.
Assume that you obtain N = 625 subjects and calculate both a sample
mean of 90 and an unbiased sample variance of 16, i.e., M = 90,
$\mathrm{s}^{2}$ = $\hat{\sigma}^{2}$ = 16. There are enough
subjects to use our large sample result concerning $\mathbf{s}^{2}$
as an estimate of $\sigma^{2}$.
\bigskip
a) Under the assumptions of normality for the underlying test
scores, what are 95\% and 99\% confidence intervals for the mean
$\mu$? What could you say if we could not assume normality of the
test scores?
\smallskip
b) With how many $\sigma$ units of the true mean will the sample
estimate fall with probability .95 (Hays, pp.\ 256--257)
\smallskip
c) Using the 95\% (99\%) confidence interval in (a), what simple
``null'' hypotheses regarding $\mu$ could you reject at .05 (.01)?
\smallskip
d) To what group of children do your results generalize? (Hint:
What type of sampling did we perform?)
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3) Subjective ratings on a 1 to 10 scale given to the ``goodness''
of the simple figure (given below) by 34 people have a frequency
table that is also given below:
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\begin{tabular}{cc}
Rating & Number of People \\ \hline
1 & 1 \\
2 & 4 \\
3 & 2 \\
4 & 11 \\
5 & 5 \\
6 & 4 \\
7 & 4 \\
8 & 3 \\
9 & 0 \\
10 & 0 \\ \hline
& N = 34 \\
\end{tabular}
\bigskip
Under the assumption that the mean rating has a normal distribution
and we can estimate $\sigma^{2}$ precisely enough to use the large
sample estimate of $\mathrm{s}^{2}$, what is a 95\% confidence
interval for the ``true'' mean $\mu$? Would you reject the
hypothesis that the figure is given the average rating of 5.5?
Provide the p-value.
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4) For the two figures given below, we have the responses of 34
individuals.
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\begin{tabular}{cccccc}
Person & & & Person & & \\ \hline
1 & 4 & 5 & 18 & 4 & 10 \\
2 & 4 & 7 & 19 & 7 & 8 \\
3 & 4 & 5 & 20 & 7 & 8 \\
4 & 4 & 4 & 21 & 5 & 8 \\
5 & 6 & 8 & 22 & 8 & 7 \\
6 & 3 & 5 & 23 & 4 & 5 \\
7 & 5 & 9 & 24 & 8 & 9 \\
8 & 4 & 7 & 25 & 6 & 9 \\
9 & 7 & 10 & 26 & 2 & 8 \\
10 &4 & 9 & 27 & 8 & 10 \\
11 & 6 & 7 & 28 & 4 & 9 \\
12 & 7 & 10 & 29 & 2 & 7 \\
13 & 5 & 7 & 30 & 7 & 9 \\
14 & 2 & 9 & 31 & 3 & 5 \\
15 & 6 & 9 & 32 & 7 & 5 \\
16 & 8 & 1 & 33 & 3 & 6 \\
17 & 4 & 7 & 34 & 5 & 8 \\
& & & Total & 173 & 250 \\
\end{tabular}
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a) Perform a paired t-test of the difference between the mean
ratings at $\alpha$ = .05. Construct the corresponding 95\%
confidence interval for the true mean difference.
\smallskip
b) Assume (incorrectly, of course) that we actually obtained these
ratings from two independent samples of subjects. Perform the
appropriate t-test of the difference between mean ratings at
$\alpha$ = .05. Construct the corresponding 95\% confidence
interval for the true mean difference. Are the standard
deviations in the two groups comparable? How do the results based
on this test compare with the paired t-test in (a)? Can you
explain any difference? Include a discussion of the reduction in
the degrees of freedom and the differing denominators for the two
test statistics.
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