Psychology 406 Assignment #3

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

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

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

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

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

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

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

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

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?

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).

e) What is E(X + Y)?

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).

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

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

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

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

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

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).

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).

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

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

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

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

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

c) Can you calculate the variance without additional information?

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