. Wildlife biologists inspect 153 deer taken by hunters and find 32 of them carrying ticks that test positive for Lyme disease.
a) Construct and interpret a 90% confidence interval for the percentage of deer that may carry such ticks. Show all work and formulas.
b) If the wildlife biologists would like to cut the margin of error found in part (a) to 2.5%, how many deer must they inspect? (Keeping the same level of confidence)
c) What concerns, if any, do you have about how the sample the wildlife biologists used was collected?
a) To construct a confidence interval for the percentage of deer carrying ticks that test positive for Lyme disease, we can use the formula:
CI = p̂ ± Zα/2 * √(p̂(1-p̂)/n)
Where:
CI is the confidence interval
p̂ is the sample proportion (32/153)
Zα/2 is the critical value for a 90% confidence level
n is the sample size (153)
First, let's calculate p̂:
p̂ = 32/153 ≈ 0.209
Next, we need to find the critical value Zα/2. For a 90% confidence level, the alpha level (α) would be 1 - 0.90 = 0.10. Dividing that by 2 gives us an alpha level of 0.05 for each tail. We can look up this value in a standard normal distribution table or use a calculator to find that Z0.05/2 is approximately 1.645.
Now, we can plug in the values into the formula:
CI = 0.209 ± 1.645 * √((0.209 * (1 - 0.209))/153)
Calculating the expression within the square root, we get:
CI = 0.209 ± 1.645 * √(0.1651/153)
CI = 0.209 ± 1.645 * 0.0258
CI ≈ 0.209 ± 0.0424
Therefore, the 90% confidence interval for the percentage of deer carrying ticks that test positive for Lyme disease is approximately 16.6% to 25.2%.
b) To find out how many deer the wildlife biologists must inspect to achieve a margin of error of 2.5%, we can use the formula:
n = (Zα/2)^2 * p̂(1-p̂) / E^2
Where:
n is the required sample size
Zα/2 is the critical value for a 90% confidence level (1.645)
p̂ is the estimated sample proportion (0.209)
E is the desired margin of error (0.025)
Plugging in the values:
n = (1.645)^2 * 0.209(1 - 0.209) / (0.025)^2
n ≈ 350.42
Therefore, they would need to inspect at least 351 deer to achieve a margin of error of 2.5% while maintaining the same level of confidence.
c) Some concerns about how the sample the wildlife biologists used was collected could be:
- Were the deer randomly selected?
- Did the sample adequately represent the population of deer?
- Was the sample size large enough for reliable results?
- Was there any bias in how the deer were chosen or how the ticks were identified?
- Were any important variables not considered in the sampling process?
Addressing these concerns is important to ensure the sample is representative of the population and that the conclusions drawn from the study are valid.