For this problem, carry at least four digits after the decimal in your calculati
ID: 2922501 • Letter: F
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 26 out of 854 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 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.
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.
(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.
No; np > 5 and nq < 5.
Yes; np < 5 and nq < 5.
lower limit upper limitExplanation / Answer
TRADITIONAL METHOD
given that,
possibile chances (x)=26
sample size(n)=854
success rate ( p )= x/n = 0.0304
I.
sample proportion = 0.0304
standard error = Sqrt ( (0.0304*0.9696) /854) )
= 0.0059
II.B9
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.01
from standard normal table, two tailed z /2 =2.576
margin of error = 2.576 * 0.0059
= 0.0151
III.
CI = [ p ± margin of error ]
confidence interval = [0.0304 ± 0.0151]
= [ 0.0153 , 0.0456]
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DIRECT METHOD
given that,
possibile chances (x)=26
sample size(n)=854
success rate ( p )= x/n = 0.0304
CI = confidence interval
confidence interval = [ 0.0304 ± 2.576 * Sqrt ( (0.0304*0.9696) /854) ) ]
= [0.0304 - 2.576 * Sqrt ( (0.0304*0.9696) /854) , 0.0304 + 2.576 * Sqrt ( (0.0304*0.9696) /854) ]
= [0.0153 , 0.0456]
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interpretations:
1. We are 99% sure that the interval [ 0.0153 , 0.0456] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 99% of these intervals will contains the true population proportion
[ANSWERS]
a. ( p )= x/n = 0.0304
b. [0.0153 , 0.0456]
c. 99% of the confidence intervals created using this method would include
the true catch-and-release mortality rate.
d. n*p>5, 854*0.0304> 5 => 25.9616>5
n*(1-p)>5, 854*0.9696> 5 => 25.9616>5
can use normal approximation
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