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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:
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\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}
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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:
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\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}
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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).
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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).
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c) What is the standard deviation of the twenty scores observed for
X?
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d) What is the standardized score for an observation of +3; -1?
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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}
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a) What are the mean, median, and mode?
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b) Is the distribution skewed? In what direction? Is the
distribution symmetric?
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c) Can you calculate the variance without additional information?
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