Question

Suppose you analyzed the ages listed on marriage licenses in Cumberland County, Pennsylvania for some period. You examine 100 licenses for your study. You find that the bride was younger than the groom for 67 couples, and the groom was younger than the bride for 27 couples. Both people listed the same age on the marriage license for the remaining 6 couples, so disregard them and consider 94 as the sample size.

Activity 6 Answers:

a) Determine a 99% confidence interval for the proportion of all marriages in this county for which the bride is younger than the groom.

b) Determine a 99% confidence interval for the proportion of all marriages in this county for which the groom is younger than the bride.

c) Comment on how the interval in part b compares to the interval in part a.

d) Comment on whether the sample data suggest that the bride is younger than the groom for more than half of the marriages in this county.

Answer 1

a)

Level of Significance,   α =    0.01          
Number of Items of Interest,   x =   67          
Sample Size,   n =    94          
                  
Sample Proportion ,    p̂ = x/n =    0.7128          
z -value =   Zα/2 =    2.576   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.046669          
margin of error , E = Z*SE =    2.576   *   0.04667   =   0.1202
                  
99%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.71277   -   0.12021   =   0.59255
Interval Upper Limit = p̂ + E =   0.71277   +   0.12021   =   0.83298
                  
99%   confidence interval is (   0.593   < p <    0.833   )

b)

Level of Significance,   α =    0.01          
Number of Items of Interest,   x =   27          
Sample Size,   n =    94          
                  
Sample Proportion ,    p̂ = x/n =    0.2872          
z -value =   Zα/2 =    2.576   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.046669          
margin of error , E = Z*SE =    2.576   *   0.04667   =   0.1202
                  
99%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.28723   -   0.12021   =   0.16702
Interval Upper Limit = p̂ + E =   0.28723   +   0.12021   =   0.40745
                  
99%   confidence interval is (   0.167   < p <    0.407   )

c) interval width in both parts is same

however interval in part a) is hifted to right of interval b)

d)

the sample data suggest that the bride is younger than the groom for more than half of the marriages in this county because both ends of confidence interval is greater than 0.50