In: Statistics and Probability
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 815 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.
We are 99% confident that the true catch-and-release mortality rate falls within this interval.
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.
(c) Is the normal approximation to the binomial justified in this
problem? Explain.
No; np < 5 and nq > 5.
No; np > 5 and nq < 5.
Yes; np > 5 and nq > 5.
Yes; np < 5 and nq < 5.
Solution :
Given that,
n = 815
x = 32
a) Point estimate = sample proportion = = x / n = 32 / 815 = 0.0393
1 - = 1 - 0.0393 = 0.9607
b) At 99% confidence level
= 1 - 99%
=1 - 0.99 =0.01
/2
= 0.005
Z/2
= Z0.005 = 2.576
Margin of error = E = Z / 2 * (( * (1 - )) / n)
= 2.576 (((0.0393 * 0.9607) / 815)
= 0.0175
A 99% confidence interval for population proportion p is ,
± E
= 0.0393 ± 0.0175
= ( 0.022, 0.057 )
lower limit = 0.022
upper limit = 0.057
We are 99% confident that the true catch-and-release mortality rate falls within this interval.
c) Yes; np > 5 and nq > 5