Question

In a survey of 2 comma 418 ​adults, 1 comma 886 reported that​ e-mails are easy to​ misinterpret, but only 1 comma 228 reported that telephone conversations are easy to misinterpret. Complete parts​ (a) through​ (c) below.

a. Construct a​ 95% confidence interval estimate for the population proportion of adults who report that​ e-mails are easy to misinterpret. less than or equalspiless than or equals ​(Round to four decimal places as​ needed.)

b. Construct a​ 95% confidence interval estimate for the population proportion of adults who report that telephone conversations are easy to misinterpret. less than or equalspiless than or equals ​(Round to four decimal places as​ needed.)

c. Compare the results of​ (a) and​ (b). Which statement below regarding the implications of the information found in​ (a) and​ (b) is​ correct?

A. More adults believe that​ e-mails are easy to misinterpret than believe that telephone conversations are easy to misinterpret.

B. The number of adults that believe that​ e-mails are easy to misinterpret and the number of adults that believe that telephone conversations are easy to misinterpret are roughly the same.

C. More adults believe that telephone conversations are easy to misinterpret than believe that​ e-mails are easy to misinterpret.

D. The information cannot be compared because it is derived from two different opinions.

Answer 1

a)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   886          
Sample Size,   n =    2418          
                  
Sample Proportion ,    p̂ = x/n =    0.366          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0098          
margin of error , E = Z*SE =    1.960   *   0.0098   =   0.0192
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.366   -   0.0192   =   0.3472
Interval Upper Limit = p̂ + E =   0.366   +   0.0192   =   0.3856
                  
95%   confidence interval is (   0.3472   < p <    0.3856   )

b)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   1228          
Sample Size,   n =    2418          
                  
Sample Proportion ,    p̂ = x/n =    0.508          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0102          
margin of error , E = Z*SE =    1.960   *   0.0102   =   0.0199
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.508   -   0.0199   =   0.4879
Interval Upper Limit = p̂ + E =   0.508   +   0.0199   =   0.5278
                  
95%   confidence interval is (   0.4879   < p <    0.5278   )

c)

C. More adults believe that telephone conversations are easy to misinterpret than believe that​ e-mails are easy to misinterpret.