In August 2002 47 of parents who had children in grades K12

In August 2002, 47% of parents who had children in grades K-12 were satisfied with the quality of education the students receive. In September 2014 the Gallup organisation conducted a poll of 1013 parents who have children in grades K-12 and asked if they were satisfied with the quality of education the students receive. Of the 1013 surveyed 437 indicated they were satisfied. Is there evidence at the alpha = 0.05 significance level that parents\' feelings have changed? Based on the sample, construct a 95% confidence interval for the percent of who are satisfied. Based on this interval, is there evidence parents feelings have changed?

Solution

a)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   =   0.47
Ha:   p   =/=   0.47
As we see, the hypothesized po =   0.47      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.431391905      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.015681303      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -2.462046378      
          
As this is a    2   tailed test, then, getting the p value,  
          
p =    0.01381468      
significance level =    0.05      

As P < 0.05, we   REJECT THE NULL HYPOTHESIS.

Thus, there is significant evidence that the proportion of parents satisfied with the quality of education is now different from 47%, at 0.05 level. [CONCLUSION]

**********************

b)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.431391905          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.01556101          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.03049902          
lower bound = p^ - z(alpha/2) * sp =   0.400892885          
upper bound = p^ + z(alpha/2) * sp =    0.461890925          
              
Thus, the confidence interval is              
              
(   0.400892885   ,   0.461890925   )

As 0.47 is not inside this interval, yes, there is evidence that parents feelings have changed. [CONCLUSION]