TC 310
MODES OF REASONING
D. Hamermesh
Fall 2002
Answer Key, First Midterm, September 26, 2002
1.
(12 minutes)
Jane wants to meet someone new and is looking through the
Austin
Chronicle
personal ads.
She is picky about who she goes out with, and is particularly
concerned about height.
Her ideal date is someone who is within one inch of her height,
which is 67’’ (i.e., she wants someone between 66 and 68 inches).
Here are the heights
of men advertising in the
Chronicle
.
Height:
71
73
72
70
72
69
70
68
68
69
71
70
67
a.
Calculate the mean and standard deviation for this sample of men.
Calculate the
coefficient of variation. Calculate the median too.
Mean = (sum of observations)/(number of observations) = 980/13 = 70
St. Dev= RMS of the deviations from the mean = 1.71 or 1.78
CV = sd
x
/mean of x = 1.71/70 = .024
or 1.78/70=.025
Median: 70
b.
Draw a histogram of the information.
Can you convey the information in a way
that would be especially useful to Jane?
Does the density function appear to be
well approximated by the normal?
The area under the histogram should sum to 100%.
0
.
0 05
.
0 1
.
0 15
.
0 2
.
0 25
The distribution looks normal.
It is symmetric around the mean.
However, there are so
few observations that even one more at the ends would skew the distribution.
2.
(12 minutes)
a.
Many of the most intriguing ads don’t provide a height.
What are Jane’s chances
of meeting an ideal date if she responds to an ad without a reported height?
Use
the mean and standard deviation above and assume the distribution is normal.
Z1 [Pr(HEIGHT)<68] = (6870)/1.71 = .2594
Pr(x<Z1) = .1211
Z2 [Pr(HEIGHT)<66] = (6670)/1.71 = 2.3392
Pr(x<Z2) = .0097
.1211  .0097 =
.1114
or 11.14%
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentb.
What crucial assumption did you need to make about the heights of men who
don’t report their heights compared with those who do?
Is this assumption
realistic?
How does this influence your estimate of Jane’s probability of getting
an ideal date?
You need to assume that the heights of those who don’t report are distributed the same
(with the same mean and standard deviation) as those who report their height – i.e. that
the reported heights are a true random sample of men who put ads in.
If men assume that
being taller is preferable to being short, then the mean of height of the reported sample
may be higher than the mean of height for the entire population of men placing ads. It’s
difficult to tell how this influences Jane’s likelihood of getting her ideal height.
Since her
ideal is below the mean, if the population mean were somewhat less she would have more
chance of getting lucky.
If the mean was much lower than her preferences, however, she
might even have less chance.
If men thought being short was preferable, than the height
reported would underestimate true height, and Jane would have less probability of an
ideal date.
Also, the standard deviation of the sample is likely to be greater than for the selfreported
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 HAMERMESH
 Normal Distribution, Standard Deviation, Mean, Parkinson, Capitol 10K

Click to edit the document details