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        Psychology 406

        Assignment \#7


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1)  Suppose that a sample of size 8 is randomly chosen from a
normally distributed population with mean 24 and variance 9.  The
standardized value corresponding to each observed value is computed
by subtracting 24 and dividing the result by 3.  Let T denote the
sum of the squares of the 8 standardized values, and find:

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a) P(T $\ge$ 20.09)

b) P(2.73 $<$ T $<$ 5.07)

c) P(T $>$ 2.18)

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2) Suppose that the daily changes in the Dow-Jones Average of
Industrial Stocks are normally distributed and that the change on
any given day is independent of the change on any other day.  A
random sample of 81 daily changes is obtained, with sample mean .20
and sample variance ($\mathbf{S}^{2}$) 1.50.  Suppose also that a
second sample is obtained, with a sample size of 25, a sample mean
of .15, and a sample variance of 1.20 ($\mathbf{S}^{2}$).  Find a
90\% confidence interval for $\sigma^{2}$ based on

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a) the first sample only

b) the second sample only

c) both samples (use the ``pooling'' formula and the approximation
in Hays, p.\ 359, for large degrees of freedom)

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3) An IQ test is given to a randomly selected group of 10 Freshman
at a given university and also to a randomly selected group of 5
Seniors at the same university. For the Freshman, the sample mean is
120 and the sample variance ($\mathrm{s}^{2}$) is 196.  For the
Seniors, the sample mean is 128 and the sample variance is 121
($\mathrm{s}^{2}$).  Find a 95\% confidence interval for the ratio
of the variance of the Freshman population to the variance of the
Senior population, e.g., $\frac{\sigma_{F}^{2}}{\sigma_{S}^{2}}$ .
Do you reject equality of the two variances at $\alpha$ = .05?




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