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 24 out of 851 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.

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

1% of all confidence intervals 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.

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.

Yes; np > 5 and nq > 5.

No; np > 5 and nq < 5.

No; np < 5 and nq > 5.

Yes; np < 5 and nq < 5.

Answer 1

Solution :

Given that,

n = 851

x = 24

a) Point estimate = sample proportion = = x / n = 24 / 851 = 0.0282

1 - = 1 - 0.0282 = 0.9718

b) At 99% confidence level the z is,

= 1 - 99%

= 1 - 0.99 = 0.01

/2 = 0.005

Z_/2 = 2.576

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.576 * (((0.0282 * 0.9718) / 851)

= 0.015

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.0282 - 0.015 < p < 0.0282 + 0.015

(0.013 < p < 0.043)

lower limit = 0.013

upper limit = 0.043

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

c) Yes; np > 5 and nq > 5.