Answers
1.
a. z = (83-70)/6=2.17
b. z = (66-70)/6 = -0.67
c. 95.25%
d. 1.5%
e. 90.82%
f. 79.96 ... 1.66*6 = 9.96
....
70 + 9.96 = 79.96
g. 56.98...-2.17 * 6 = -13.02
....
70 - 13.02 = 56.98
h. This observation is 1.64 sd above the mean, which is 1.64 * 6 = 9.84 points, so the score is 79.84
points.
i. This observation is 0.77 sd below the mean. (Note 22%ile rank means that 28 % of the observations
are between this score and the mean, so you look up .2800 in the body of the table.) So 0.77 * 6 = 4.62
points below the mean. The score is 70 – 4.62 = 65.48
j. The score 60 and 80 are both, conveniently, 10 points below the mean, making this a symmetric
problem. You can find the area 10 points above and multiply by 2. The z score for 10 points above the
mean is 10/6 = 1.67, which includes 45.25 % of cases. So the total is 45.25% * 2 = 90.5%
k. The most extreme 20 percent of cases would be the 10 percent at the bottom end and the 10 percent at
the top end, again making this a symmetric problem. The cutoff for the10% tail includes the 40% area,
so the associated z score for the value of .4000 is 1.28. Converting this z score to a real score gives us
7.68 units above and below the mean. So the cutoffs are 70 - 7.68 = 62.32 and 77.68.

2.
Area from 45 to 52
Z = (45-52)/ - 8.3 =
- 0.84 ... This transforms to 29.95%
Area from 52 to 55
Z = (55-52)/8.3 = 0.36 ... This transforms to 14.06%
Together, this means 44.01 percent of House districts were competitive.
3.
a.
Z = (3.5 – 3.2)/1.1 = 0.3/1.1 =
0.27
b .
Using table, proportion above z + 0.27 is
0.3936.
So 1.000 – 0.3936 =
0.6064.
So, with rounding, percentile rank is 60.6%
c.
z = (2.1 – 3.2)/1.1
= -1.1/1.1 = -1.0

4.
a.
Z = (700 – 650)/88 =
50/88 =
0.57
Using table, the percentage above this Z is 28.43 %
b.
Z score for 600 is same as Z score for 700, but negative.
Thus, we know from part a of this problem
that
28.43% lies outside of (below) 600 and 28.43 % lies outside of (above) 700, for a total of 56.86
percent outside these two points.
Thus, 100%
- 56.86% = 43.14% is the answer.
c.
Use table to find a number in the body of the table close to 25% or 0.2500.
Closest is 0.2514,
corresponding to a Z score of -0.67.
Plug this into Z formula
-0.67 = (x – 650)/88
.
Solving for x, we get 591.

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- Summer '17
- Wimpy
- Normal Distribution, Standard Deviation, Percentile rank, Standard score, Democratic Campaign Committee