Using the binomial​ distribution, 

would 

you be surprised​ if, after observing 

1010 

​individuals, fewer than half covered their mouth when​ sneezing? Why?

▼
 
Yes,

No,
 
it 

▼
 
would

would not
 
be​ surprising, because the probability is 

enter your response here​, 

which is 

▼
 
greater than

less than
 
0.05.

​(Round to four decimal places as​ needed.)

To determine if it would be surprising for fewer than half of 1010 individuals to cover their mouth when sneezing, we need to use the binomial distribution and calculate the probability of that event.

### Step-by-Step Solution:

1. **Define the variables**:
   - Let \( X \) be the number of individuals who cover their mouth when sneezing.
   - Total number of individuals: \( n = 1010 \).
   - Probability of success (covering mouth): \( p \).
     - Since we are not provided with \( p \), let's assume \( p = 0.5 \) (assuming a fair 50% probability that a person covers their mouth).

2. **Set up the binomial distribution**:
   The binomial distribution formula for the probability of observing exactly \( k \) successes (covering their mouth) in \( n \) trials is:
   \[
   P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
   \]

3. **Determine the range of interest**:
   - We are interested in the probability of observing fewer than half of the individuals covering their mouth, so \( X < 505 \) (since \( \frac{1010}{2} = 505 \)).

4. **Approximate using the normal distribution**:
   For large \( n \), the binomial distribution can be approximated by a normal distribution with:
   - Mean \( \mu = np = 1010 \times 0.5 = 505 \)
   - Standard deviation \( \sigma = \sqrt{np(1 - p)} = \sqrt{1010 \times 0.5 \times 0.5} \approx 15.91 \)

   Now, we calculate the Z-score for \( X = 505 \):
   \[
   Z = \frac{505 - 505}{15.91} = 0
   \]

5. **Calculate the probability**:
   We are looking for \( P(X < 505) \), which corresponds to \( P(Z < 0) \) in the normal distribution. From the standard normal table, we know that \( P(Z < 0) = 0.5 \).

Since the probability is \( 0.5 \), which is **greater than 0.05**, it would **not** be surprising for fewer than half of the individuals to cover their mouth when sneezing.

### Final Answer:
- It **would not** be surprising, because the probability is 0.5, which is **greater than 0.05**.

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:
1. How would the result change if the probability \( p \) of covering the mouth was not 0.5?
2. What is the significance of using the normal approximation in this case?
3. How can the binomial distribution be used to model other real-world events?
4. What happens to the probability if we increase the sample size from 1010 to 5000?
5. Can you calculate the exact binomial probability without using the normal approximation?

**Tip**: Normal approximations work well for large sample sizes, but for smaller sample sizes, it's often better to use the binomial formula directly.

Using the binomial distribution, would you be surprised if, after observing 1010 individuals, fewer than half covered their mouth when sneezing? Why? (Round to four decimal places as needed.)

Explore the use of the binomial distribution and normal approximation to determine if it would be surprising for fewer than half of 1010 individuals to cover their mouth when sneezing. Learn about key concepts such as probability, normal distribution, and the Central Limit Theorem. Suitable for Grades 11-12 and college-level students.

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