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chapter10_exercise.lyx
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chapter10_exercise.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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theorems-ams
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figs-within-sections
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\end_header
\begin_body
\begin_layout Part*
Chapter 10.
Categorical Data And Nonparametric Methods
\end_layout
\begin_layout Chapter*
10.1 Tests of Goodness-of-Fit
\end_layout
\begin_layout Section*
Summary
\end_layout
\begin_layout Standard
The
\begin_inset Formula $\chi^{2}$
\end_inset
test of goodness-of-fit was introduced as a method for testing the null
hypothesis that our data form an i.i.d.
sample from a specific distribution against the alternative hypothesis
that the data have some other distribution.
The test is most natural when the specific distribution is discrete.
Suppose that there are
\begin_inset Formula $k$
\end_inset
possible values for each observation, and we observe
\begin_inset Formula $N_{i}$
\end_inset
with value
\begin_inset Formula $i$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
.
Suppose that the null hypothesis says that the probability of the
\begin_inset Formula $i$
\end_inset
th possible value is
\begin_inset Formula $p_{i}^{0}$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
.
Then we compute
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
Q=\sum_{i=1}^{k}\frac{(N_{i}-np_{i}^{0})^{2}}{np_{i}^{0}},
\]
\end_inset
\end_layout
\begin_layout Standard
where
\begin_inset Formula $n=\sum_{i=1}^{k}N_{i}$
\end_inset
is the sample size.
When the null hypothesis says that the data have a continuous distribution,
then one must first create a corresponding discrete distribution.
One does this by dividing the real line into finitely many (say,
\begin_inset Formula $k$
\end_inset
) intervals, calculating the probabiliy of each interval
\begin_inset Formula $p_{1}^{0},...,p_{k}^{0}$
\end_inset
, and then preteding as if all we learned from the data were into which
intervals each observation fell.
This converts the original data into discrete data with
\begin_inset Formula $k$
\end_inset
possible values.For example, the value of
\begin_inset Formula $N_{i}$
\end_inset
used in the formula for
\begin_inset Formula $Q$
\end_inset
is number of observations that fell into
\begin_inset Formula $i$
\end_inset
th interval.
All of the
\begin_inset Formula $\chi^{2}$
\end_inset
test statistics in this text have the form
\begin_inset Formula $\sum\text{(observed-expected)}^{2}/\text{expected },$
\end_inset
where
\begin_inset Quotes eld
\end_inset
observed
\begin_inset Quotes erd
\end_inset
stands for an observed count and expeted stands for the expected value
of the observed count under the assumption that the null hypothesis is
true.
\end_layout
\begin_layout Section*
Exercises
\end_layout
\begin_layout Exercise*
1.
Consider the hypotheses being tested in Example 10.1.5.
Use a test procedure of the form outlined in Exercise 11 of Sec 9.1 and
compare the result to the numerical result obtained in Example 10.1.5.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\bar{X}_{n}=\sum_{i=1}^{100}X_{i}/n\sim N(10,\sigma/\sqrt{N})$
\end_inset
,
\begin_inset Formula $\bar{X}_{n}=0.16$
\end_inset
.
so find
\begin_inset Formula $z$
\end_inset
, then find
\begin_inset Formula $Z\ge|z|$
\end_inset
, is
\begin_inset Formula $p-value$
\end_inset
\end_layout
\begin_layout Exercise*
2.
Show that if
\begin_inset Formula $p_{i}^{0}=1/k$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
then the statistic
\begin_inset Formula $Q$
\end_inset
defined by Eq.
(10.1.2) can be written in the form
\end_layout
\begin_layout Exercise*
\begin_inset Formula
\[
Q=(\frac{k}{n}\sum_{i=1}^{k}N_{i}^{2})-n
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Standard
In Eq.
(10.1.2),
\begin_inset Formula $Q=\sum_{i=1}^{k}\frac{(N_{i}-np_{i}^{0})^{2}}{np_{i}^{0}}$
\end_inset
.
Instituting
\begin_inset Formula $p_{i}^{0}=1/k$
\end_inset
, we will collect the result.
\end_layout
\begin_layout Exercise*
3.
Investigate the
\begin_inset Quotes eld
\end_inset
randomness
\begin_inset Quotes erd
\end_inset
of your favorite pseudo-random numbers generator as follows.
Simulate
\begin_inset Formula $200$
\end_inset
psuedo-random numbers between
\begin_inset Formula $0$
\end_inset
and
\begin_inset Formula $1$
\end_inset
and divide the unit interval into
\begin_inset Formula $k=10$
\end_inset
intervals of length
\begin_inset Formula $0.1$
\end_inset
each.
Apply the
\begin_inset Formula $\chi^{2}$
\end_inset
test of the hypothesis that each of the
\begin_inset Formula $10$
\end_inset
intervals has the same probability of containing a pseudo-random number.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $H_{0}:p_{i}=0.1\forall i$
\end_inset
.
\begin_inset Formula $H_{1}:p_{i}\neq0.1\exists i$
\end_inset
.
\end_layout
\begin_layout Standard
Using randomness generator program to randomize number and count in these
intervals, we can collect
\begin_inset Formula $N_{i}$
\end_inset
.
and then apply
\begin_inset Formula $\chi^{2}$
\end_inset
test.
\end_layout
\begin_layout Exercise*
4.
Find
\begin_inset Formula $p$
\end_inset
test of
\begin_inset Formula $Q$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Exercise*
7.
Find
\begin_inset Formula $c$
\end_inset
of level of significance 0.001, then find
\begin_inset Formula $Q$
\end_inset
and compare with
\begin_inset Formula $c$
\end_inset
, we will have conclusion.s
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Exercise*
8.
Same exercise 7.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Exercise*
9.
Use standard normal distribution, and divide 5 interval with 0.2 probability,
then we can find these pivot based on proabbility and standard normal distribut
ion.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Chapter*
10.3.
Contigency Tables
\end_layout
\begin_layout Exercise*
1.
First I convert original table to
\begin_inset Formula $N_{i+}N_{+j}$
\end_inset
table.
\end_layout
\begin_layout Exercise*
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
117
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
60
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
50
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
227
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
130
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
30
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
91
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
251
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
247
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
90
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
141
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
478
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
Then we convet this table to
\begin_inset Formula $\hat{E}_{i+,+j}$
\end_inset
table
\end_layout
\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
117.3
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
42.74
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
66.96
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
129.7
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
47.26
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
74
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
Then following Eq(10.3.5),
\begin_inset Formula $Q=21.47$
\end_inset
, is
\begin_inset Formula $(3-1)(2-1)=2$
\end_inset
\begin_inset Formula $\chi^{2}$
\end_inset
distribution,
\begin_inset Formula $p-$
\end_inset
value =
\begin_inset Formula $1-Q\chi_{2}(21.47)$
\end_inset
,
\begin_inset Formula $Q\chi_{2}$
\end_inset
is cdf of
\begin_inset Formula $\chi_{2}$
\end_inset
\end_layout
\begin_layout Exercise*
10.
Solving with this problem with two-way contingency tables
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Exercise*
5.
Corresponding tail area lies between (0.025, 0.05) is real number is in (7.815,9.34
8).
\series bold
Not relevent to
\begin_inset Formula $\chi_{k-s-1}$
\end_inset
and
\begin_inset Formula $\chi_{k-1}$
\end_inset
on testing based on MLE based on sample set.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Chapter*
10.5 Simpson's paradox
\end_layout
\begin_layout Exercise*
1.
Summarization of problem :
\end_layout
\begin_layout Exercise*
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Men (%)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Characteristic
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Not Characteristc
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
I
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
80
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
20
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
II
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
60
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
40
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Exercise*
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Women(%)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Characteristic
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Not Characteristic
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
I
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
30
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
70
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
II
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
10
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
90
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset