Question

The Consumer Reports National Research Center conducted a telephone survey of 2,000 adults to learn about the major economic concerns for the future. The survey results showed that 1,640 of the respondents think the future health of Social Security is a major economic concern. If computing the confidence intervals manually, make sure to use at least three decimal digits for the critical values

. a. What is the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern?

b. At 90% confidence, what is the margin of error (to 4 decimals)?

c. Develop a 90% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern (to 3 decimals). ( , )

d. Develop a 95% confidence interval for this population proportion (to 3 decimals). ( , )

Answer 1

a)
p^ = point estimate of the population proportion

= x / n

= 1640/2000

=0.82

b)
              
we get the standard error of p, sp
              
sp = sqrt[p^ (1 - p^) / n]

= 0.008591
              
Now, for the critical z,alpha/2 =   0.05          
Thus, z(alpha/2) =    1.64485   

E = z(alpha/2) * sp

= 0.014131

c)   
              
lower bound = p^ - z(alpha/2) * sp =   0.805869568          
upper bound = p^ + z(alpha/2) * sp =    0.834130432          
              
Thus, the confidence interval is              
              
CI=( 0.805869568 , 0.834130432 )

d)

p^ = point estimate of the population proportion

= x / n

=0.82          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n]

= 0.008590693          
              
Now, for the critical z,              
alpha/2 = 0.025          
z(alpha/2) =1.959963985          
Thus,              
lower bound = p^ - z(alpha/2) * sp =   0.803162552          
upper bound = p^ + z(alpha/2) * sp =    0.836837448          
              
Thus, the confidence interval is              
              
CI=(0.803162552 , 0.836837448)

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