Question

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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 857 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. O We are 1% confident that the true catch-and-release mortality rate falls above this interval. We are 1% confident that the true catch-and-release mortality rate falls within this interval. We are 99% confident that the true catch-and-release mortality rate falls outside this interval. We are 99% confident that the true catch-and-release mortality rate falls within this interval. (c) Is the normal approximation to the binomial justified in this problem? Explain. No; np > 5 and ng < 5. Yes; np < 5 and ng < 5. No; np < 5 and ng > 5. O Yes; np > 5 and ng > 5.

Answer 1

a) a point estimate of p is $\hat{p} = \frac{28}{857} \approx 0.0327$

b)

Let p denotes the population proportion. Let o denotes the sample proportion and n denotes the sample size. Here, 28 857 р 0.033 * (1 – 0.033) 857 0.0327, n = 857, standard error = 0.006073 0.0061 At 99 percent confidence level, Margin of error is *(1 - ) ME= 0.033* (1 – 0.033) 857 * 2 1-0.99 2 * 20.005 0.033 * (1 – 0.033) 857 * 2.575829 = 0.015642 0.016 n a 99 percent confidence interval for the population proportion p is © * (1 - i) = + * 2 1-0.99 2 ጎኔ = 0.0327 + 0.033 * (1 – 0.033) 857 * 20.005 = 0.0327 + 0.033* (1 – 0.033) 857 * 2.575829 = 0.0327 +0.015642 = (0.017030, 0.048314) ~ (0.017, 0.048)

Lower limit = 0.017

Upper limit = 0.048

We are 99% confident that the true catch-and-release mortality rate falls within this interval.

c) Yes, n*p > 5 and n*q > 5