ST3239: Survey Methodology - The Department of Statistics and ...

Solution
A. The proportions choosing “banned” are independent of each other; a high value does not
force a low value of the other. Thus, an appropriate estimate of this difference is
√
0.44 × 0.56
0.44 − 0.08 ± 2
+
600
0.08 × 0.92
200
= 0.36 ± 0.06
B. The proportion of nonsmokers choosing “special areas” is dependent on the proportions
choosing “banned”; if the latter is large, the former must be small. These are multinomial
proportions. Thus, an appropriate estimate of this difference is
√
0.44 × 0.56
0.52 − 0.44 ± 2
+
600
0.52 × 0.48
600
+ 2 ×
0.44 × 0.52
600
= 0.08 ± 0.08
Example The major league baseball season in US came to an abrupt end in the middle of
1994. In a poll of 600 adult Americans, 29% blamed the players for the strike, 34% blamed
the owners, and the rest held various other opinions. Does evidence suggest that the true
proportions who blame players and owner, respectively, are really different?
p 1 : proportions of Americans who blamed the players.
p 2 : proportions of Americans who blamed the owners.
ˆV ar(ˆp 1 − ˆp 2 ) = ˆp 1ˆq 1
n + ˆp 2ˆq 2
n + 2ˆp 1ˆp 2
n
= ˆ0.29 × ˆ0.71 0.34 × 0.66
+
600 600
= 1.0458 × 10 −3
So an approximate 95% C.I. for p 1 − p 2 is
+
0.29 − 0.34 ± z 0.025
√
ˆV ar(ˆp1 − ˆp 2 )
= −0.05 ± 1.96 × 0.03234
= (−0.11339, 0.01339)
2 × 0.29 × 0.34
600
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