the percentage of high school graduates applying to multiple colleges has been i

Question

the percentage of high school graduates applying to multiple colleges has been increasing over the past several years. According to the National Association for College Admission Counseling 29% of high school students applied to seven or more colleges in 2011. A random sample of 350 high school graduates in 2013 was selected and 84 of them applied to seven or more colleges. A) Construct a 90% confidence interval to estimate the actual proportion of high school graduates that applied to seven or more colleges in 2013. B) What is the margin of error for this example? C) is there evidence that the proportion has changed since 2011 based on this sample?

Explanation / Answer

A)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.24          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.022828554          
              
Now, for the critical z,              
alpha/2 =   0.05          
Thus, z(alpha/2) =    1.644853627          
Thus,              
      
lower bound = p^ - z(alpha/2) * sp =   0.202450371          
upper bound = p^ + z(alpha/2) * sp =    0.277549629          
              
Thus, the confidence interval is              
              
(   0.202450371   ,   0.277549629   ) [ANSWER]

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b)

Margin of error = z(alpha/2)*sp =    0.037549629   [ANSWER]

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c)

YES, as this whole interval does not include 0.29 inside it. [ANSWER]

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