Corn: In a random sample of 84 ears of corn, farmer Carl finds that 9 of them ha

Question

Corn: In a random sample of 84 ears of corn, farmer Carl finds that 9 of them have worms. He wants to find the 99% confidence interval for the proportion of all his corn that has worms.

(a) What is the point estimate for the proportion of all of Carl's corn that has worms? Round your answer to 3 decimal places.


(b) What is the critical value of z (denoted z/2) for a 99% confidence interval? Use the value from the table or, if using software, round to 2 decimal places.
z/2 =  

(c) What is the margin of error (E) for a 99% confidence interval? Round your answer to 3 decimal places.
E =  

(d) Construct the 99% confidence interval for the proportion of all of Carl's corn that has worms. Round your answers to 3 decimal places.
< p <   

(e) Based on your answer to part (d), are you 99% confident that less than 22% of Carl's corn has worms?

a. Yes, because 0.22 is below the upper limit of the confidence interval.

b. No, because 0.22 is above the upper limit of the confidence interval.      

c. No, because 0.22 is below the upper limit of the confidence interval.

d. Yes, because 0.22 is above the upper limit of the confidence interval.

Explanation / Answer

A)

The point estimate is the sample proportion,
              
p^ = point estimate of the population proportion = x / n = 9/84 =   0.107142857   [ANSWER]

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b)      

Now, for the critical z,              

alpha/2 =   0.005          

Thus, z(alpha/2) =    2.575829304 = 2.58   [ANSWER]

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c)

Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.033746828          
              
Now, for the critical z,              

alpha/2 =   0.005          

Thus, z(alpha/2) =    2.58          

Thus,              

Margin of error = z(alpha/2)*sp =    0.087066816 [ANSWER]

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d)
          
lower bound = p^ - z(alpha/2) * sp =   0.020076041          
upper bound = p^ + z(alpha/2) * sp =    0.194209673          
              
Thus, the confidence interval is              
              
(   0.020076041   ,   0.194209673   ) [ANSWER]

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e)

OPTION D: d. Yes, because 0.22 is above the upper limit of the confidence interval. [ANSWER]

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