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Homework Assignment #3 - Solutions
1. Social Network Friends
The following is a data set from the DVD attached to the book, supposedly corresponding to the “Try It Yourself”
question on page 304, even though the data is completely different.
Consider this a simple random sample of 30 members of a social network, listing their friends in the network:
249
227
215
214
201
196
179
172
162
156
146
145
137
137
135
131
130
127
125
124
114
108
1.1. Descriptive Statistics
Compute the following for the sample:
• Sample Mean
• Population and Sample Variance
• Population and Sample Standard Deviation
• Median
• Suggest your choice for 1st and 3rd quartiles
Solutions
Mean
Median
1 quartile
3 quartile
Pop. S.D.
Pop. Variance
Standard Deviation
Sample Variance
138.5
133
102
169.5
50.1881
2518.85
51.0461
2605.71
100
95
87
80
74
71
63
55
1.2. Confidence Intervals
Compute confidence intervals, at a level of your choice, for
• The “true” mean (note that the “true” variance is unknown)
• The “true” variance (note that the “true” mean is unknown)
Note 1: The two estimates are not independent, since they share the same data. ¡It is incorrect
to
¢
combine them in some simple “2-dimensional confidence interval” for the pair ¹; ¾ 2
Note 2: Both estimates require the assumption that the data is coming from a normal
distribution. This is a bit of a tall order, however it happens that applying a typical nonparametric test for normality to this data set does not rule out that possibility (the cautious
language should become more familiar once we go into the chapter on statistical tests)
Note 3: You could probably work out the required statistics from the raw data, given that you
have time, but, just in case, here are our customary summaries:
Count
30
Sum
4,155
Sum of Squares
651,033
Solutions
For the mean:
90% CI for the Mean from
to
122.665
154.335
95% CI for the Mean from
to
119.439
157.561
99% CI for the Mean from
to
112.811
164.189
For the variance:
levels
0.9
0.95
0.99
left
right
1775.632
4267.220
1652.706
4708.99
1443.864
5759.061
The numerator in the chi-square formula,
75565.5
X
2
2
(xi ¡ x¹ ) = nS = (n ¡ 1)s2 , is
2. Body Temperatures
This is a reduced sample of readings of body temperatures. Consider it a simple random sample
of size 32:
99.6
99.4
99.2
98.9
98.9
98.7
98.7
98.6
2.1.
98.6
98.6
98.6
98.6
98.5
98.4
98.4
98.4
98.4
98.3
98
98
98
98
97.9
97.8
Descriptive Statistics
Compute the following for the sample:
• Sample Mean
• Population and Sample Variance
• Population and Sample Standard Deviation
• Median
• Suggest your choice for 1st and 3rd quartiles
Solutions
Mean
Median
1 quartile
3 quartile
Pop. S.D.
Pop. Variance
Standard Deviation
Sample Variance
98.2875
98.4
97.8
98.6
0.56555
0.31984
0.5746
0.33016
97.8
97.7
97.6
97.6
97.6
97.6
97.4
97.4
2.2. Confidence Intervals
Compute confidence intervals, at a level of your choice, for
• The “true” mean (note that the “true” variance is unknown)
• The “true” variance (note that the “true” mean is unknown)
Note 1: The two estimates are not independent, since they share the same data. ¡It is incorrect
to
¢
combine them in some simple “2-dimensional confidence interval” for the pair ¹; ¾ 2
Note 2: Both estimates require the assumption that the data is coming from a normal
distribution. This is a bit of a tall order, however it happens that applying a typical nonparametric test for normality to this data set does not rule out that possibility (the cautious
language should become more familiar once we go into the chapter on statistical tests)
Note 3: You could probably work out the required statistics from the raw data, given that you
have time, but, just in case, here are our customary summaries:
Count
Sum
Sum of Squares
32
3,145.2
309,144.08
Solutions
90% CI for the Mean from
to
98.1153
98.4597
95% CI for the Mean from
to
98.0803
98.4947
99% CI for the Mean from
98.0088
to
98.5662
For the variance, the numerator (n ¡ 1)s2 turns out to be
10.235
so that we have the confidence intervals
levels
0.9
0.95
0.99
left
right
0.186082
0.707924
0.227519
0.530845
0.212204
0.583565