2. Wildlife biologists inspect 153 deer taken by hunters and find 32 of them carrying Lyme disease ticks. d) If the scientists want to cut the margin of error, E in half, calculate the number of deer that must be inspected? Consider the same 98% C.I. Show all mathematics.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![**Wildlife Biology Study on Lyme Disease Ticks**
In a recent study conducted by wildlife biologists, 153 deer taken by hunters were inspected, and 32 of these deer were found to be carrying Lyme disease ticks.
**Cutting the Margin of Error in Half**
**Problem Statement:**
If the scientists want to cut the margin of error (\(E\)) in half, calculate the number of deer that must be inspected. Consider the same 98% Confidence Interval (C.I.). Show all mathematics.
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For educational purposes, let us go through the steps to determine the required sample size when the margin of error is halved, keeping the confidence level constant at 98%.
### Step-by-Step Solution:
**Step 1: Initial conditions and formula for margin of error**
The margin of error (\(E\)) for a proportion can be defined as:
\[ E = z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Where:
- \( z \) is the z-value for the desired confidence level
- \( \hat{p} \) is the sample proportion (in this case, \(\hat{p} = \frac{32}{153}\))
- \( n \) is the sample size
**Step 2: Finding the sample proportion**
\[ \hat{p} = \frac{32}{153} \approx 0.209 \]
**Step 3: Determining the z-value for a 98% Confidence Interval**
From standard z-tables, the z-value for a 98% confidence level is approximately 2.33.
**Step 4: Writing the initial margin of error equation (E)**
\[ E = 2.33 \times \sqrt{\frac{0.209 \times (1 - 0.209)}{153}} \]
Calculating the value inside the square root:
\[ 0.209 \times (1 - 0.209) = 0.209 \times 0.791 \approx 0.165 \]
Then,
\[ E = 2.33 \times \sqrt{\frac{0.165}{153}} \approx 2.33 \times \sqrt{0.00108} \approx 2.33 \times 0.0328 \approx 0.076 \]
**Step 5: Halving the margin of error (\(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedb1c45e-845e-4ed7-afc8-deed75acb3e2%2F8a869d9f-d0e8-4323-a4b1-147962723589%2F5dm0wg9.jpeg&w=3840&q=75)
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