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 32 out of 860 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.)

lower limit:

upper limit:

Give a brief explanation of the meaning of the interval. (choose the correct)

a. 1% of all confidence intervals would include the true catch-and-release mortality rate.

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

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

d. 99% of all confidence intervals would include the true catch-and-release mortality rate.

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

a. Yes; np > 5 and nq > 5.

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

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

Yes; np < 5 and nq < 5.

Answer 1

a)

sample proportion, = 0.0372

b)

sample size, n = 860
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.0372 * (1 - 0.0372)/860) = 0.0065

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

Margin of Error, ME = zc * SE
ME = 2.58 * 0.0065
ME = 0.0168

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.0372 - 2.58 * 0.0065 , 0.0372 + 2.58 * 0.0065)
CI = (0.02 , 0.054)
Lower limit = 0.020
Upper limit = 0.054

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

c)

a. Yes; np > 5 and nq > 5.