wildlife biologists inspect 165 deer taken by hunters and find 34 or 20.6% of them carry ticks that test positive for Lyme disease. Calculate the 99% margin of error and confidence interval for the proportion of deer that may carry ticks. How many deer are needed to be inspected if the margin of error were to be cut in half? Comment on the statistical concerns about this study if it were reported in the news.

Use a confidence interval formula for proportions. Here's one:

CI99 = p + or - (2.58)(√pq/n)
...where p = .206, q = 1 - p, and n = 165.

Plug the values into the formula and calculate.

I hope this will help get you started on the first part of your problem.

To calculate the margin of error and confidence interval, we need to use the following formula:

Margin of Error = Critical Value * Standard Error

Confidence Interval = Proportion + Margin of Error

Step 1: Calculate the Standard Error
To calculate the standard error, we use the formula:

Standard Error = sqrt((Proportion * (1 - Proportion)) / Sample Size)

In this case, the proportion of ticks that test positive for Lyme disease is 20.6% or 0.206. The sample size is 165.

Standard Error = sqrt((0.206 * (1 - 0.206)) / 165)

Step 2: Determine the Critical Value
Next, we need to determine the critical value for a 99% confidence level. The critical value corresponds to the z-score. For a 99% confidence level, the critical value is approximately 2.576.

Step 3: Calculate the Margin of Error
Now we can calculate the margin of error by multiplying the critical value by the standard error:

Margin of Error = 2.576 * Standard Error

Step 4: Calculate the Confidence Interval
To calculate the confidence interval, we add and subtract the margin of error from the proportion:

Confidence Interval = Proportion +/- Margin of Error

Step 5: Calculate the needed sample size
To calculate the sample size needed to cut the margin of error in half, we can use the formula:

New Sample Size = (Sample Size / Current Margin of Error) * Desired Margin of Error

In this case, the current margin of error is the one we already calculated, and the desired margin of error is half the current margin of error.

Now let's calculate:

Step 1:
Standard Error = sqrt((0.206 * (1 - 0.206)) / 165) ≈ 0.032

Step 2:
Critical Value = 2.576

Step 3:
Margin of Error = 2.576 * 0.032 ≈ 0.0827

Step 4:
Confidence Interval = 0.206 +/- 0.0827 ≈ (0.123, 0.289)

Step 5:
New Sample Size = (165 / 0.0827) * (0.0827 / 2) ≈ 332.64

Therefore, approximately 333 deer would need to be inspected if the margin of error were cut in half.

Statistical Concerns:
If this study were reported in the news, there are some statistical concerns to consider. Firstly, the sample size of 165 deer may not be representative of the entire population, which can introduce sampling bias. Additionally, the study only focused on deer taken by hunters, which might not reflect the entire deer population accurately. Furthermore, factors such as geographic location, time of year, and other variables that could influence the prevalence of ticks and Lyme disease were not mentioned. Lastly, it is crucial to interpret the results with caution and understand that the confidence interval does not guarantee the exact proportion of deer with ticks in the population.