For this problem, carry at least four digits after the decimal in your calculati

Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 28 out of 836 fish died when caught and released using barbless hooks on flies or lures. All hooks were removed from the fish.

(a) Let p represent the proportion of all pike and trout that die (i.e., p is the mortality rate) when caught and released using barbless hooks. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answers to three decimal places.)

Give a brief explanation of the meaning of the interval.

99% of all confidence intervals would include the true catch-and-release mortality rate. 1% of all confidence intervals would include the true catch-and-release mortality rate.     1% of the confidence intervals created using this method would include the true catch-and-release mortality rate. 99% of the confidence intervals created using this method would include the true catch-and-release mortality rate.

(c) Is the normal approximation to the binomial justified in this problem? Explain.

No; np < 5 and nq > 5. Yes; np > 5 and nq > 5.     Yes; np < 5 and nq < 5. No; np > 5 and nq < 5.

Explanation / Answer

a)

Note that              
              
p^ = point estimate of the population proportion = x / n = 28/836 =   0.033492823   [ANSWER]

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b)      
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.006222649          
              
Now, for the critical z,              
alpha/2 =   0.005          
Thus, z(alpha/2) =    2.575829304          
Thus,              
Margin of error = z(alpha/2)*sp =    0.016028481          
lower bound = p^ - z(alpha/2) * sp =   0.017464342          
upper bound = p^ + z(alpha/2) * sp =    0.049521304          
              
Thus, the confidence interval is              
              
(   0.017464342   ,   0.049521304   ) [ANSWER]

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OPTION D: 99% of the confidence intervals created using this method would include the true catch-and-release mortality rate. [ANSWER]

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OPTION B: Yes; np > 5 and nq > 5. [ANSWER]

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