1. The way a question is framed in a survey can have an impact on reponses. The
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1. The way a question is framed in a survey can have an impact on reponses. The Pew Research Center conducted a survey with the following question By 2014, nearly all Americans will be required to have health insurance. People who do not buy insurance will have to pay a penalty) while lpeople who cannot afford it will recieve help from the government. Do you approve or disapprove this law? For each randomly sampled respondent, the statements in brackets were randomized, such that each person either heard the part about penalities or they heard the part about government help (but not both). Here are the results: Sample size Approve law Disapprove law Government help 47 53 732 Penalties 37 63 (a) Construct a 90% confidence interval for the difference in approval (b) Interpret your confidence interval.Explanation / Answer
The 100(1-) % confidence interval for difference in population proportion (p1-p2) is given by:
(p1 - p2 ) ± z(1-/2) [ p1(1- p1)/n1 + p2(1- p2)/n2)]
Where p1 is the proportion of first sample = x1/n1
p2 is the proportion of second sample = x2/n2
n1 and n2 are the sample sizes of the two samples.
z(1-/2) is the reliability coefficient taken from z table.
a)p1 = 47/771 = 0.06096
p2 = 37/732 = 0.05055
The 90% confidence interval is given by:
(0.06096 – 0.05055) ± 1.645* 0.012632
Where = 1 – (90/100) = 0.10
Critical probability = 1- /2 = 0.95
From the z table, z score having cumulative probability equal to 0.95 = 1.645
The 90% confidence interval:
(0.01041 ± 0.02078) = (-0.01037, 0.03119)
b) Interpretation: The true difference is reasonably anywhere from 3.1% more approval of government help to 1.03% more approval of penalties.
c) H0 = There is no significant difference in population proportion, that is, p1 = p2
H1 = There is significant difference in population proportion, that is, p1 p2
z = |p1 - p2 | / [ p1(1- p1)/n1 + p2(1- p2)/n2)]
Where p1 is the proportion of first sample = x1/n1
p2 is the proportion of second sample = x2/n2
n1 and n2 are the sample sizes of the two samples.
p1 = 53/771 = 0.0687419
p2 = 63/732 = 0.0860656
z = |-0.017324|/0.0137721
z = 1.2579
z/2 = 1.645
Since, z/2=1.645 > zcal = 1.2579, we accept the null hypothesis and hence conclude that there is no significant difference in population proportion.
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