93 A statistics practitioner working for Major League Baseba

9.3. A statistics practitioner working for Major League Baseball wants to supply radio and television commentators with interesting statistics. He observed several hundred games and counted the number of times a runner on first base attempted to steal second base. He found there were 373 such events, of which 259 were successful. Estimate with 95% confidence the proportion of all attempted thefts of second base that are successful.

Solution

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.694369973          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.023852783          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.046750596          
lower bound = p^ - z(alpha/2) * sp =   0.647619377          
upper bound = p^ + z(alpha/2) * sp =    0.741120569          
              
Thus, the confidence interval is              
              
(   0.647619377   ,   0.741120569   ) [ANSWER]