According to the Monitoring the Future Project in 2009 624 o

According to the Monitoring the Future Project in 2009, 62.4% of 12th graders felt that there is a great risk to a persons health if he or she has 4 or 5 drinks nearly every day. In 2010, a random sample of 2440 12th graders is taken this year, and 1491 think that there is a great risk to a persons health if he or she has 4 or 5 drinks nearly every day.

a. [4 0 ] Find a 95% Wald interval for population proportion in 2010 of 12th graders who think that there is a great rick to a person’s health if he or she has 4 or 5 drinks nearly everyday.

b.  Using your confidence interval from part (a), is there evidence that the proportion in 2010 of 12th graders who think that there is a great rick to a person’s health if he or she has 4 or 5 drinks nearly everyday decreased since 2009? Explain.

Solution

a)

First, we get the point estimate of the proportion, p^,              
              
p^ = x / n =    0.611065574          
              
Here, a previous estimte for the population proportion is

p = 0.624

Hence, we get the standard error of p, sp:              
              
sp = sqrt[p(1 - p) / n] =    0.009805987          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
              
lower bound = p^ - z(alpha/2) * sp   0.591846193          
upper bound = p^ + z(alpha/2) * sp =    0.630284955          
              
Thus, the confidence interval is              
              
(   0.591846193   ,   0.630284955   ) [ANSWER]

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b)

As 0.624 is still inside this confidence interval, then NO, THERE IS NO SIGNIFICANT EVIDENCE the proportion in 2010 of 12th graders who think that there is a great risk to a person’s health if he or she has 4 or 5 drinks nearly everyday decreased since 2009. [CONCLUSION]