Question

In a telephone survey completed in the Spring of 2014, a randomly selected number of USadults were asked the question “Is there solid evidence that the earth has been warming?”They were also asked their political party preference. The following is a partial summary of the results.

1. What is the sample proportion for adults who prefer the Republican party who wouldanswer “Yes” to the question?

   What is the approximate standard error for the sampling distribution of sample proportions for this group of Republicans?

   What is the margin of error for a 95% confidence interval?

   What is a 95% confidence interval for the percent of all adults who prefer the Republican party who would answer “Yes” to the question?

2. What is the sample proportion for adults who prefer the Democratic party who wouldanswer “Yes” to the question?

   What is the approximate standard error for the sampling distribution of sample proportions for this group of Democrats?

   What is the margin of error for a 95% confidence interval?

	Republican	Democratic
Yes, solid evidence	154	363
No, no solid evidence	307	171

What is a 95% confidence interval for the percent of all adults who prefer the Democratic party who would answer “Yes” to the question?

c. Looking at separate confidence intervals is generally not a good method for making a conclusion. For this part you will compute a 95% confidence interval for the difference in the population proportions for Republicans and Democrats, by completing the following steps:

Calculate the sample difference in proportions, ?̂? − ?̂?.
Calculate the approximate standard error, s, of the sampling distribution for differences in

sample proportions.

Calculate a 99% confidence interval for the difference in the population proportions,?? −??.

In terms a non-statistics person would understand, interpret your 99% confidence interval, explaining what it tells us about the proportions of Republicans and Democratswho would answer “Yes” to the question.

Answer 1

Number of Items of Interest,   x =   154
Sample Size,   n =    517
      
Sample Proportion ,    p̂ = x/n =    0.2979

Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0201          
margin of error , E = Z*SE =    1.960   *   0.0201   =   0.0394

95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.298   -   0.0394   =   0.2585
Interval Upper Limit = p̂ + E =   0.298   +   0.0394   =   0.3373
                  
95%   confidence interval is (   0.2585   < p <    0.3373   )

-----------------------------

Number of Items of Interest,   x =   363
Sample Size,   n =    517
      
Sample Proportion ,    p̂ = x/n =    0.7021

Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0201          
margin of error , E = Z*SE =    1.960   *   0.0201   =   0.0394
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.702   -   0.0394   =   0.6627
Interval Upper Limit = p̂ + E =   0.702   +   0.0394   =   0.7415
                  
95%   confidence interval is (   0.6627   < p <    0.7415   )

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first sample size,     n1=   517          
number of successes, sample 1 =     x1=   154          
proportion success of sample 1 , p̂1=   x1/n1=   0.2979          
                  
sample #2   ----->   standard          
second sample size,     n2 =    517          
number of successes, sample 2 =     x2 =    363          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.702          
                  
difference in sample proportions, p̂1 - p̂2 =     0.2979   -   0.7021   =   -0.4043

Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0284          
margin of error , E = Z*SE =    2.576   *   0.0284   =   0.0733
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.404   -   0.0733   =   -0.4775
upper limit = (p̂1 - p̂2) + E =    -0.404   +   0.0733   =   -0.3310
                  
so, confidence interval is (   -0.4775   < p1 - p2 <   -0.3310   )  

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