4. Comparing two population proportions (independent samples) Aa Aa Most major s

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4. Comparing two population proportions (independent samples) Aa Aa Most major survey research organizations do not include wireless telephone numbers when conducting random-digit-dial telephone surveys. If there are differences between persons with and without landline phones using a random-digit-dial telephone survey may introduce bias into the survey results. Data on a broad range of health topics are collected through personal household interviews of a representative sample of the U.S. civilian noninstitutionalized population in the National Health Interview Survey (NHIS). Since respondents are also asked about their telephones, the NHIS data allow you to compare the health insurance status of persons with and without landline phones. Let pi denote the proportion of adults living in wireless phone-only homes who are uninsured and p2 denote the proportion of adults living in landline phone homes who are uninsured. Independent random samples are selected from the two populations. Sample 1, with sample size n1 = 69, provides a sample proportion pja 0.29. Sample 2, with sample size n2 = 128, provides a sample proportion p78 0.14. (The sample results are representative of the data collected in the NHIS, but the sample sizes are much smaller.) Use the Distributions tool to help you answer the questions that follow. Standard Normal Distribution Mean = 0 Standard Deviation = 1 -3 -2 -1 0 2

Explanation / Answer

4.

a.
TRADITIONAL METHOD
given that,
sample one, x1 =20.01, n1 =69, p1= x1/n1=0.29
sample two, x2 =17.92, n2 =128, p2= x2/n2=0.14
I.
standard error = sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where
p1, p2 = proportion of both sample observation
n1, n2 = sample size
standard error = sqrt( (0.29*0.71/69) +(0.14 * 0.86/128))
=0.063
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
margin of error = 1.96 * 0.063
=0.123
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.29-0.14) ±0.123]
= [ 0.027 , 0.273]
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DIRECT METHOD
given that,
sample one, x1 =20.01, n1 =69, p1= x1/n1=0.29
sample two, x2 =17.92, n2 =128, p2= x2/n2=0.14
CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.29-0.14) ± 1.96 * 0.063]
= [ 0.027 , 0.273 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 95% sure that the interval [ 0.027 , 0.273] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the difference between
true population mean P1-P2

we are 95% sure that the interval [ 0.027 , 0.273] contains the difference between
true population proportion P1-P2, we conclude that being uninsured is more prevalent among adults living in wireless phone only homes compared to adults living in landline phone homes

b.
Given that,
sample one, x1 =20.01, n1 =69, p1= x1/n1=0.29
sample two, x2 =17.92, n2 =128, p2= x2/n2=0.14
null, Ho: p1= p2
alternate, H1: p1 > p2
level of significance, = 0.05
from standard normal table,right tailed z /2 =1.64
since our test is right-tailed
reject Ho, if zo > 1.64
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.29-0.14)/sqrt((0.193*0.807(1/69+1/128))
zo =2.547
| zo | =2.547
critical value
the value of |z | at los 0.05% is 1.64
we got |zo| =2.547 & | z | =1.64
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: right tail - Ha : ( p > 2.5472 ) = 0.00543
hence value of p0.05 > 0.00543,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 > p2
test statistic: 2.547
critical value: 1.64
decision: reject Ho
p-value: 0.00543
we conclude that being uninsured is more prevalent among adults living in wireless phone only homes compared to adults living in landline phone homes

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