A study considered the question, \"Are you a registered voter?\" Accuracy of res

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 87 people were asked the voter registration question face to face. Of those sampled, seventy-five respondents gave accurate answers (as verified by city records). Another random sample of 91 people were asked the same question during a telephone interview. Of those sampled, sixty-eight respondents gave accurate answers. Assume the samples are representative of the general population. Categorize the problem below according to parameter being estimated, proportion p, mean mu, difference of means mu_1 - mu_2, or difference of proportions p_1 - p_2. Then solve the problem. Let p_1 be the population proportion of all people who answer the voter registration question accurately during a face-to-face interview. Let P_2 be the population proportion of all people who answer the question accurately during a telephone interview. Find a 95% confidence interval for p_1 - p_2. (Use 3 decimal places.) 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. We can not make any conclusions using this confidence interval. Because the interval contains only negative numbers, we can say there is a higher proportion of accurate responses in telephone interviews.

Explanation / Answer

a)

As we talk about two proportions,

OPTION A: p1 - p2

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b)

Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.862068966      
p2 = x2/n2 =    0.747252747      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.058670161      
          
          
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 =    0.114991402      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    -0.000175184      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    0.229807621      
          
Thus, the confidence interval is          
          
(   -0.000175184   ,   0.229807621 ) [ANSWER]

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c)

It is mixed (although the lower bound cannot be detected at 3 decimal places).

Hence,

OPTION B: 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. [ANSWER]

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Hi! Part c is a very close call, because -0.000175 cannot be detected at 3 decimal places. Still, we have the exact answer, so B is still the best answer. The second best answer is option A.

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