5. Inferences about the difference between two population proportions Aa Aa Most

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5. Inferences about the difference between two population proportions 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 status of persons with and without landline phones Let pi denote the proportion of adults living in wireless phone-only homes who are obese and p2 denote the proportion of adults living in landline phone homes who are obese. Independent random samples are selected from the two populations. Sample 1, with sample size n1-515, provides a sample proportion 1-0.23. Sample 2, with sample size n2 - 760, provides a sample proportion p2 0.28. (The sample results are representative of the data collected in the NHIS, but the sample sizes are much smaller.)

Explanation / Answer

1.

Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.23      
p2 = x2/n2 =    0.28      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.024680897      
          
For the   95%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.025      
z(alpha/2) =    1.959963985      
Margin of error = z(alpha/2)*sd =    0.048373668      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    -0.098373668      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    -0.001626332      
          
Thus, the confidence interval is          
          
(   -0.098373668   ,   -0.001626332 )   

= (-0.098, -0.002) [ANSWER]

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As we can see, the whole interval is < 0, so, p1 < p2, then

BAsed on the 95% confidence....

Blank 1: [LESS]
Blank 2: [UNDERESTIMATE]

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Under the assumption that the null hypothesis is true....

Blank 1: As above, sd = 0.024680897 = 0.0250 [ANSWER]
Blank 2: z = [p1 - p2 - pdo]/sd =    -2 [ANSWER]
Blank 3: P =    0.0228 [ANSWER]
Blank 4: As P < 0.05, CAN. [ANSWER]

      

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