I need a help to answer this question! thank you! A sociologist takes a random s

Question

I need a help to answer this question! thank you!

A sociologist takes a random sample of 300 students from the 27,000 students enrolledat a large state university. There were 222 undergraduates and 78 graduate students in thesample.

a) If p = the proportion of undergraduates among the entire student population at the state university, compute ˆp and ˆq.

b) What assumptions and conditions must be considered before we construct acondence interval? Verify that the conditions are satised.

c) Construct a 95% condence interval for p.

d) What is the margin of error?

e) In the same quarter, a second sociologist also takes a random sample of 300students from this university and uses the same procedure to calculate a 95% condenceinterval. True or false, and explain fully: the second sociologist will arrive at exactly the same condence interval as the rst one.

Explanation / Answer

a)

p^ = x/n = 222/300 = 0.74 [ANSWER]

q^ = 1 - p^ = 0.26 [ANSWER]

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b)

It has to be that n < 0.05N. As n = 300, N = 27000, then this is satisfied.

Also, there are more than 10 of each of undergraduates and graduates, so np^ > 10 and nq^ > 10, so that is satisfied as well.

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c)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.74          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.02532456          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
      
lower bound = p^ - z(alpha/2) * sp =   0.690364775          
upper bound = p^ + z(alpha/2) * sp =    0.789635225          
              
Thus, the confidence interval is              
              
(   0.690364775   ,   0.789635225   ) [ANSWER]

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d)

Margin of error = z(alpha/2)*sp =    0.049635225   [ANSWER]

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e)

FALSE. There is some variation in p^, so once the second experimenter gets a different p^ (highly likely), then we will get a diffrent confidence interval.

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