Question

A study considered the question, 'Are you a registered voter?" Accuracy of response was confirmed by a check of city voting records. Two methods of survey were used: a face-to-face interview and a telephone interview. A random sample of 85 people were asked the voter registration question face to face. Of those sampled, seventy-six respondents gave accurate answers (as verified by city records). Another random sample of 90 people were asked the same question during a telephone interview. Of those sampled, sixty-seven respondents gave accurate answers. Assume the samples are representative of the general population. (a) Categorize the problem below according to parameter being estimated, proportion P, mean , difference of means 41 - 42, or difference of proportions Pi-P2. Then solve the problem. OH1-42 OP1 - P2 (b) Let Pi be the population proportion of all people who answer the voter registration question accurately during a face-to-face interview. Let P2 be the population proportion of all people who answer the question accurately during a telephone interview. Find a 95% confidence interval for P1 - P2. (Use 3 decimal places.) lower limit upper limit (c) Does the interval contain numbers that are all positive? all negative? mixed? Comment on the meaning of the confidence interval in the context of this problem. At the 95% level, do you detect any difference in the proportion of accurate responses from face-to-face interviews compared with the proportion of accurate responses from telephone interviews? Because the interval contains only positive numbers, we can say that there is a higher proportion of accurate responses in face-to-face interviews. Because the interval contains both positive and negative numbers, we can not say that there is a higher proportion of accurate responses in face-to-face interviews. We can not make any conclusions using this confidence interval. Because the interval contains only negative numbers, we can say that there is a higher proportion of accurate responses in telephone interviews.

Answer 1

TRADITIONAL METHOD
given that,
sample one, x1 =76, n1 =85, p1= x1/n1=0.894
sample two, x2 =67, n2 =90, p2= x2/n2=0.744
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( (0.894*0.106/85) +(0.744 * 0.256/90))
=0.057
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
margin of error = 1.96 * 0.057
=0.111
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.894-0.744) ±0.111]
= [ 0.038 , 0.261]
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DIRECT METHOD
given that,
sample one, x1 =76, n1 =85, p1= x1/n1=0.894
sample two, x2 =67, n2 =90, p2= x2/n2=0.744
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 = [ (0.894-0.744) ± 1.96 * 0.057]
= [ 0.038 , 0.261 ]
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interpretations:
1) we are 95% sure that the interval [ 0.038 , 0.261] 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
Answers:
a.
difference of proportions p1-p2
option:B
b.
95% sure that the interval [ 0.038 , 0.261]
c.
option:A
because the interval contains only positive numbers ,we can say that there is a higher proportion
of accurate responses in face to face interviews.