Question

Suppose a veterinarian wants to estimate the difference between the proportion of cat owners who are single and the proportion of dog owners who are single. Of the pet owners that visit any of the veterinarian clinics in her city regularly, she identifies 3872 pet owners that exclusively have cats and 4108 pet owners that exclusively have dogs. From this list of pet owners, she surveys 149 randomly selected cat owners and 126 randomly selected dog owners and asks each of them if they are single or married. Her findings are summarized in the table. Population Population description Sample size Number of successes Sample proportion 1 cat owners ?1=149 ?1=68 ?̂ 1=0.45638 2 dog owners ?2=126 ?2=54 ?̂ 2=0.42857 Calculate the upper and lower limits (bounds) for a large sample 99% ?‑confidence interval for the difference in two population proportions, ?1−?2. Give each of your answers with three decimal places of precision. Then, complete the following sentence to state the interpretation of the confidence interval. The that the who are single is between and .

Answer 1

first sample size,     n1=   149
number of successes, sample 1 =     x1=   68
proportion success of sample 1 , p̂1=   x1/n1=   0.45638

      
second sample size,     n2 =    126
number of successes, sample 2 =     x2 =    54
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.42857

difference in sample proportions,    p̂1 - p̂2 =     0.02780441

Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0601  
Z critical value =   Z α/2 =    2.5758   [excel function: =normsinv(α/2)
          
margin of error , E =   Z*SE =    0.1547  
          
confidence interval is           
lower limit =    (p̂1 - p̂2) - E =    -0.1269  
upper limit =   (p̂1 - p̂2) + E =    0.1825  

we are 99% confident that difference between the proportion of cat owners who are single and the proportion of dog owners who are single is betwwen ( - 0.127 , 0.183)