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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?)

\bigskip

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.

 \bigskip

 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 \\

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 \bigskip

 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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