Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 140 eligible voters aged​ 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged​ 18-24, 22% of them voted.
Probability that exactly 37 voted

To find the probability that exactly 37 out of 140 eligible voters aged 18-24 voted using a normal approximation, we can follow these steps:

### Step 1: Define the problem
We have:
- \( n = 140 \): the number of trials (voters selected)
- \( p = 0.22 \): the probability of success (voting)

We want to find the probability that exactly 37 voters voted, i.e., \( P(X = 37) \).

### Step 2: Check the conditions for normal approximation
For the normal approximation to the binomial distribution to be valid, both \( np \) and \( n(1-p) \) should be greater than 5:
- \( np = 140 \times 0.22 = 30.8 \)
- \( n(1-p) = 140 \times 0.78 = 109.2 \)

Both values are greater than 5, so the normal approximation is appropriate.

### Step 3: Find the mean and standard deviation of the binomial distribution
- Mean \( \mu = np = 140 \times 0.22 = 30.8 \)
- Standard deviation \( \sigma = \sqrt{np(1-p)} = \sqrt{140 \times 0.22 \times 0.78} \approx 4.80 \)

### Step 4: Apply the continuity correction
We need to apply a continuity correction because we are using a continuous distribution to approximate a discrete one. For \( P(X = 37) \), we look at the interval from \( 36.5 \) to \( 37.5 \).

### Step 5: Convert to the standard normal distribution
We convert the binomial problem to a standard normal distribution problem using the \( Z \)-score formula:

\[ Z = \frac{X - \mu}{\sigma} \]

For \( X = 36.5 \):
\[ Z_1 = \frac{36.5 - 30.8}{4.80} \approx 1.19 \]

For \( X = 37.5 \):
\[ Z_2 = \frac{37.5 - 30.8}{4.80} \approx 1.40 \]

### Step 6: Find the probabilities
Using the standard normal distribution table (or a calculator), we find the probabilities corresponding to \( Z_1 \) and \( Z_2 \):

\[ P(Z < 1.19) \approx 0.8830 \]
\[ P(Z < 1.40) \approx 0.9192 \]

### Step 7: Calculate the desired probability
The probability that exactly 37 voters voted is the difference between these two probabilities:

\[ P(36.5 < X < 37.5) = P(Z < 1.40) - P(Z < 1.19) \]
\[ P(36.5 < X < 37.5) \approx 0.9192 - 0.8830 = 0.0362 \]

So, the probability that exactly 37 out of 140 eligible voters aged 18-24 voted is approximately 0.0362, or 3.62%.

Would you like more details or have any questions?

### Related Questions:
1. What is the probability that fewer than 30 voters voted?
2. What is the probability that more than 40 voters voted?
3. How would the probability change if the voting percentage was 25% instead of 22%?
4. What is the expected number of voters who voted?
5. How does the standard deviation affect the shape of the normal curve in this context?
6. What is the Z-score for exactly 37 voters?
7. How would the probability change if 200 voters were selected instead of 140?
8. How can the normal approximation be validated against the exact binomial probability?

### Tip:
When using the normal approximation for the binomial distribution, always apply the continuity correction to improve the accuracy of the approximation.

Calculate the probability that exactly 37 out of 140 eligible voters aged 18-24 voted using a normal approximation. Understand the application of the binomial distribution, normal approximation, and central limit theorem in this scenario.

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