Question

You are interested in constructing a 99% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 392 randomly selected caterpillars observed, 49 lived to become butterflies. Round answers to 4 decimal places where possible.

a. With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between ___ and ____.

b. If many groups of 392 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About ____ percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about ____ percent will not contain the true population proportion.

Answer 1

a)

Level of Significance,   α =    0.01          
Number of Items of Interest,   x =   49          
Sample Size,   n =    392          
                  
Sample Proportion ,    p̂ = x/n =    0.125          
z -value =   Zα/2 =    2.576   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0167          
margin of error , E = Z*SE =    2.576   *   0.0167   =   0.0430
                  
99%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.125   -   0.0430   =   0.0820
Interval Upper Limit = p̂ + E =   0.125   +   0.0430   =   0.1680
                  
99%   confidence interval is (   0.0820 < p <    0.1680 )

b)

About 99% percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about 1 percent will not contain the true population proportion