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Eight leaves removed 20 82 frequency distribution among the three sex com-
TTT binations is homogeneous with respect to the dif-
26. The article “Human Lateralization from Head to ferent genotypes? Define the parameters of
Foot: Sex-Related Factors” (Science, 1978: interest, state the appropriate Ho and Hy, and
1291-1292) reports for both a sample of right- perform the analysis,
handed men and a sample of right-handed
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13.3 Two-Way Contingency Tables 753
_—_—————Gonibinalion the number of degrees of freedom for the chi-
ne Gombmadon squared statistic.
M/M M/F —FIF__ 32, Suppose that in a particular state consisting of four
distinct regions, a random sample of mn, voters is
Male ‘ 2 4 a obtained from the kth region for k = 1, 2, 3, 4.
5 af 2 a Each voter is then classified according to which
Genoty 1 ‘ 36 ‘ candidate (1, 2, or 3) he or she prefers and accord-
yenotype 5 5 MI 6 ing to voter registration (1 = Dem., 2 = Rep.,
: a és Pr 3 = Indep.). Let pj denote the proportion of
voters in region k who belong in candidate cate-
eB gory i and registration category j. The null hypoth-
29. Each individual in a random sample of high school Eds of bomogenceus, epiong ix” is
= ee DOM; POubleaeiviews 1att i tion within each candidate/registration combina-
Usage, resulting in the data displayed in the accom- tion is the same for all four regions). Assuming
paying thonway “table (“atetadks About Mari. that Ho is true, determine pj: and éj as functions
juana and Political Views,” Psych. Rep., 1973: of the observed nj,’s, and use the general rule of
1,051-1,054). Does the data support the hypothesis thumb to obtain the number of degrees of freedom
that political views and marijuana usage level are Roethejchissquared test
independent within the population? Test the appro- i
priate hypotheses using level of significance 01. 33. Consider the accompanying 2 x 3 table displaying
the sample proportions that fell in the various com-
Usage Level binations of categories (e.g., 13% of those in the
Never Rarely Frequently sample were in the first category of both factors).
Liberal 4 | a3 | no | a. Suppose the sample consisted of n = 100 peo-
Political ; ple. Use the chi-squared test for independence
Views Conservative 15 with significance level .10.
Other 1? 85 | b. Repeat ran {) assuming that the sample size
n = 1000.
¢. What is the smallest sample size n for which
30. Show that the chi-squared statistic for the test of these observed proportions would result in
independence can be written in the form rejection of the independence hypothesis?
tt yp
C=-V Vlg Io8 1 2 3
fat pat \ EG
[spe Ta]
Why is this formula more efficient computation-
ally than the defining formula for 722 2 [or | | 2 |
31. Suppose that in Exercise 29 each student had
been categorized with respect to political views, 34. Use logistic regression to test the relationship
marijuana usage, and religious preference, with between leaf removal and fruit growth in Exer-
the categories of this latter factor being Protes- cise 25. Compare the P-value with what was
tant, Catholic, and other. The data could be dis- found in Exercise 25. (Remember that } = 2°.)
played in three different two-way tables, one Explain why you expected the logistic regression
corresponding to each category of the third factor. to give a smaller P-value.
With pix = P(political category i, marijuana cat- 35° 4 random sample of 100 faculty at a university
egory j, and religious category ), the null hypoth- gives the results shown below for professorial
esis of independence of all three factors states that rank versus gender.
Pijk = Pix Pi Po Let mj denote the observed a. Test for a relationship at the 5% level using a
frequency in cell (i, j, k). Show how to estimate chi-squared statistic.
the expected cell counts assuming that Hp is true b. Test for a relationship at the 5% level using
(ix = npix, 80 the Pix’s must be determined). logistic regression.
‘Then use the general rule of thumb to determine °
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754 = cuarrer 13 Goodness-of-Fit Tests and Categorical Data Analysis
¢. Compare the P-values in parts (a) and (b). Is S EEETTEEEEEEEEEEEEEEEEEE EEE
this in accord with your expectations? Explain. Rank Male Female
d. Interpret your results. Assuming that today’s Professor 25 9
assistant professors are tomorrow’s associate “Assoc Prok 20 8
professors and professors, do you see implica- ‘Asst Prof 18 20
tions for the future? ————
Supplementary Exercises M@tsaz¥))
36. The article “Birth Order and Political Success” evidence to conclude that the proportions falling
(Psych. Rep., 1971: 1,239-1,242) reports that into the experience categories are different for
among 31 randomly selected candidates for men and women? Use 2 = .01.
political office who came from families with a
four children, 12 were firstborn, 11 were mid- Years of Experience
dleborn, and 8 were lastborn. Use this data to a. ss fa sss S50
test the null hypothesis that a political candidate Gender 13 46 79 JOR I
from such a family is equally likely to be in any Mile i © 2 we
one of the four ordinal positions. Female 230 251 238 164 258
37. The results of an experiment to assess the effect SS
of crude oil on fish parasites are described inthe 39. The authors of the article “Predicting Profes-
article “Effects of Crude Oils on the Gastrointes- sional Sports Game Outcomes from Intermediate
tinal Parasites of Two Species of Marine Fish” Game Scores” (Chance, 1992: 18-22) used a chi-
(UJ. Wildlife Diseases, 1983: 253-258). Three squared test to determine whether there was any
treatments (corresponding to populations in the merit to the idea that basketball games are not
procedure described) were compared: (1) no con- settled until the last quarter, whereas baseball
tamination, (2) contamination by 1-year-old games are over by the seventh inning. They
weathered oil, and (3) contamination by new also considered football and hockey. Data was
oil. For each treatment condition, a sample of collected for 189 basketball games, 92 baseball
fish was taken, and then each fish was classified games, 80 hockey games, and 93 football games.
as either parasitized or not parasitized. Data com- The games analyzed were sampled randomly
patible with that in the article is given. Does the from all games played during the 1990 season
data indicate that the three treatments differ with for baseball and football and for the 1990-1991
respect to the true proportion of parasitized and season for basketball and hockey. For each game,