Question

From public records, individuals were identified as having been charged with drunken driving not less than 6 months or more than 12 months from the starting date of the study. Two random samples from this group were studied. In the first sample of 30 individuals, the respondents were asked in a face-to-face interview if they had been charged with drunken driving in the last 12 months. Of these 30 people interviewed face to face, 17 answered the question accurately. The second random sample consisted of 46 people who had been charged with drunken driving. During a telephone interview, 24 of these responded accurately to the question asking if they had been charged with drunken driving during the past 12 months. Assume the samples are representative of all people recently charged with drunken driving. (a) Categorize the problem below according to parameter being estimated, proportion P, mean , difference of means H1 - Uy, or difference of proportions P1 - P2. Then solve the problem. O P1 - P2 O 41 - 112 (b) Let P1 represent the population proportion of all people with recent charges of drunken driving who respond accurately to a face-to-face interview asking if they have been charged with drunken driving during the past 12 months. Let P2 represent the population proportion of all people who respond accurately to the question when it is asked in a telephone interview. Find a 90% confidence interval for P1 - P2. (Use 3 decimal places.) lower limit upper limit (c) Does the interval found in part (a) 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 90% level, do you detect any differences in the proportion of accurate responses to the question from face-to-face interviews as 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

1)

a) p1-p2

b) sample #1   ----->  
first sample size,     n1=   30
number of successes, sample 1 =     x1=   17
proportion success of sample 1 , p̂1=   x1/n1=   0.5667
      
sample #2   ----->  
second sample size,     n2 =    46
number of successes, sample 2 =     x2 =    24
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.5217

level of significance, α =   0.10              
Z critical value =   Z α/2 =    1.645   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.1167          
margin of error , E = Z*SE =    1.645   *   0.1167   =   0.1919
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.045   -   0.1919   =   -0.147
upper limit = (p̂1 - p̂2) + E =    0.045   +   0.1919   =   0.237

c) option B)

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2)

a) p̂ = x/n =    0.4720

b) z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0393          
margin of error , E = Z*SE =    1.960   *   0.0393   =   0.0771
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.472   -   0.0771   =   0.395
Interval Upper Limit = p̂ + E =   0.472   +   0.0771   =   0.549