Please help!
In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 22 out of 845 fish died when caught and released using barbless hooks on flies or lures. All hooks were removed from the fish.
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. (Use 4 decimal places.)
Find a 90% confidence interval for p. (Use 4 decimal places.)
lower limit
upper limit
To find a point estimate and a 90% confidence interval for the mortality rate (p) of northern pike and trout when caught and released using barbless hooks, we can use the given information.
Point Estimate for p:
The point estimate for p is calculated by dividing the number of fish that died (22) by the total number of fish studied (845):
point estimate for p = 22 / 845 = 0.026 (rounded to 4 decimal places)
Confidence Interval for p:
To find a 90% confidence interval for p, we need to calculate the lower and upper limits.
Step 1: Calculate the standard error:
Standard error (SE) is the standard deviation of the sampling distribution of the proportion. It can be calculated using the following formula:
SE = sqrt((p̂ * (1 - p̂)) / n)
Where:
p̂ = point estimate for p
n = sample size (total number of fish studied)
SE = sqrt((0.026 * (1 - 0.026)) / 845) = 0.0058 (rounded to 4 decimal places)
Step 2: Calculate the margin of error:
The margin of error (ME) is calculated by multiplying the standard error by the critical value associated with the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645.
ME = 1.645 * 0.0058 = 0.0095 (rounded to 4 decimal places)
Step 3: Calculate the lower and upper limits:
The lower limit (LL) can be found by subtracting the margin of error from the point estimate, and the upper limit (UL) can be found by adding the margin of error to the point estimate.
LL = 0.026 - 0.0095 = 0.0165 (rounded to 4 decimal places)
UL = 0.026 + 0.0095 = 0.0355 (rounded to 4 decimal places)
Therefore, the 90% confidence interval for p is:
lower limit = 0.0165
upper limit = 0.0355