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The image shows a homework problem involving the binomial distribution, based on a study about sneezing habits. Let's break down the problem and solve each part step by step.

### Given:
- Probability \( p \) that a randomly selected individual will **not** cover their mouth while sneezing is \( p = 0.267 \).
- Total number of individuals observed: \( n = 12 \).

You are asked to solve three parts:

### Part (a)
**Question**: What is the probability that among 12 randomly observed individuals, exactly 8 do not cover their mouth when sneezing?

This follows a **binomial distribution**:
- \( n = 12 \)
- \( p = 0.267 \)
- We want \( P(X = 8) \).

The formula for binomial probability is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

For \( k = 8 \):

\[
P(X = 8) = \binom{12}{8} (0.267)^8 (1 - 0.267)^4
\]

### Part (b)
**Question**: What is the probability that fewer than 4 do not cover their mouth when sneezing?

This requires calculating \( P(X < 4) \), i.e., the sum of probabilities for \( X = 0, 1, 2, 3 \).

\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

Each term can be calculated using the binomial formula.

### Part (c)
**Question**: Would you be surprised if, after observing 12 individuals, fewer than half covered their mouth when sneezing? The problem specifies calculating the probability and comparing it to 0.05.

For fewer than half (fewer than 6 people) covering their mouth, you are looking for \( P(X < 6) \).

Once calculated, you can compare the result to 0.05 to decide whether it would be surprising.

---

I will calculate these values for you. Would you like detailed calculations for each part or just the final answers?

### Follow-up Questions:
1. How does the binomial distribution formula work for large \( n \) values?
2. What happens to the probability if \( p \) is increased?
3. What does it mean if a probability is less than 0.05 in hypothesis testing?
4. How would the result change if \( n \) were 20 instead of 12?
5. How can you approximate a binomial distribution with a normal distribution for large \( n \)?

**Tip**: The binomial distribution works best when trials are independent, and there are only two outcomes.

According to a study by Nick Wilson of Otago University, the probability that a randomly selected individual will not cover their mouth when sneezing is 0.267. What is the probability of exactly 8 out of 12 individuals not covering their mouth, fewer than 4 not covering, and would it be surprising for fewer than half to not cover their mouth when sneezing?

Solve binomial distribution problems based on sneezing habits, including calculating the probability of exactly 8 out of 12 not covering their mouth, fewer than 4, and determining significance for fewer than half of individuals covering their mouth. Suitable for undergraduate-level students.

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