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        Psychology 406

        Assignment \#3


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1) Hays --- Appendix A (pp. 901--911), \#24, \#26, \#32; Appendix B
(pp. 912--920), \#12, \#14, \#16


\bigskip

2) Suppose we observe the following examination scores for 30
students:

\begin{tabular}{ccccc}

 111 & 91 & 93 & 79 & 89 \\
 90   & 83  & 78  & 82 & 109  \\
 90     & 103  & 108  & 82 & 109  \\
 90       & 93  & 87 & 79 & 92 \\
 104         & 84  & 103  & 121  & 92  \\
 93           & 95 &  89 & 92  & 93 \\
\end{tabular}

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Using raw scores, what are the median, mode, and mean?

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3)  Suppose I observe a discrete random variable X that can take on
values from -4 to +4 with the following probabilities:

\bigskip

\begin{tabular}{cc}

Values of X & Probability \\

-4 & 1/36 \\
-3 & 1/18 \\
-2 & 1/9 \\
-1 & 1/12 \\
0 & 2/9 \\
+1 & 1/9 \\
+2 & 1/9 \\
+3 & 1/9 \\
+4 & 3/18 \\




\end{tabular}

\bigskip

a) What are the mean, median, and mode of X?

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b) If the values of X represent money that is won or lost, is the
``game'' represented by X a ``fair'' game?

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c) If I define a random variable Y that takes on the value 1 when X=
-4, -3, -2, -1, or 0, and takes on the value -1 when X = 1, 2, 3, or
4, what is the expected value of the number of 1's for Y in 10
observations on Y?

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d) If I multiply each value of X by 10 and add the number 5 to the
result, what is the mean of the new random variable?  (Hint: Use a
formula).

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e) What is E(X + Y)?

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f) Calculate the variance of the random variable X.  If I multiply
each value of X by 10 and add the number 5 to the result, what is
the variance of the new variable?  (Hint: Use a formula).

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g) Are X and Y independent random variables?  Show it!

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h) What is the standardized score corresponding to the value -2 for
X; corresponding to +4?

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i) What is the standard deviation of the random variable X?

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4) Assume that we have obtained 20 observations on the random
variable X in the first problem:

\bigskip

\begin{tabular}{ccccc}

-4, & +4, & -1, & -3, & +1, \\
 -1, & -3, & -2, & -2, & +2, \\
 0,  & 0, & +4, & +4, & -1, \\
 0,   & +1, & +3, & 0, & 0 \\

 \end{tabular}

 \bigskip

 a) What is the mean of these twenty observations?  If I multiply
 each of the scores by 6 and then add 2, what is the mean of the
 resultant twenty observations?  (Hint: Use a formula).

 \smallskip

 b) What is the variance of these twenty observations?  If I
 multiply each of the scores by 6 and then add 2, what is the
 variance of the resultant twenty observation?  (Hit: Use a
 formula).

 \smallskip

 c) What is the standard deviation of the twenty scores observed for
 X?

 \smallskip

 d) What is the standardized score for an observation of +3; -1?

 \bigskip

 5) Consider the density function for the continuous random variable
 X given on the following page.

\begin{figure}

\centerline{\includegraphics{density_function.eps}}


\end{figure}

\bigskip

a) What are the mean, median, and mode?

\smallskip

b) Is the distribution skewed?  In what direction?  Is the
distribution symmetric?

\smallskip

c) Can you calculate the variance without additional information?


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