A professor writes a statistics test and estimates that 90% of test takers will complete the test in 1 hour or less. She gives the test to a class of 240 students. What is the probability that 12 or less students do not complete the test in less than an hour?

This is a binomial probability problem, as the professor estimates that each test taker has a 90% chance of completing the test in less than 1 hour, and 10% of students will take longer than 1 hour. We are interested in finding the probability that 12 or fewer students out of 240 take more than 1 hour to finish.

Let's break it down:

1. **Define the binomial distribution**:
   - \( n = 240 \) (number of students)
   - \( p = 0.10 \) (probability that a student will not complete the test in under 1 hour)
   - \( X \) represents the number of students who do not complete the test in less than an hour.

   So, \( X \sim \text{Binomial}(n=240, p=0.10) \).

2. **Goal**:
   We need to calculate the probability that 12 or fewer students do not complete the test in less than 1 hour:
   \[
   P(X \leq 12)
   \]

Since \( n \) is large, we can approximate this binomial distribution using a normal distribution. For large \( n \), a binomial distribution \( \text{Binomial}(n, p) \) can be approximated by a normal distribution \( N(\mu, \sigma^2) \), where:
   - \( \mu = n \cdot p = 240 \cdot 0.10 = 24 \)
   - \( \sigma^2 = n \cdot p \cdot (1 - p) = 240 \cdot 0.10 \cdot 0.90 = 21.6 \)
   - \( \sigma = \sqrt{21.6} \approx 4.65 \)

3. **Normal approximation**:
   Using the normal approximation to the binomial, we calculate \( P(X \leq 12) \) by finding the corresponding \( z \)-score:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   Using a continuity correction, we adjust 12 to 12.5:
   \[
   z = \frac{12.5 - 24}{4.65} = \frac{-11.5}{4.65} \approx -2.47
   \]

4. **Find the probability**:
   Using standard normal distribution tables or a calculator, we find the probability corresponding to a \( z \)-score of \( -2.47 \). This gives:
   \[
   P(X \leq 12) \approx 0.0068
   \]

So, the probability that 12 or fewer students do not complete the test in less than 1 hour is approximately \( 0.0068 \) or 0.68%.

---

Would you like further clarification or details?

Here are some related questions for further exploration:
1. How would the result change if the professor expected 85% to finish in under 1 hour?
2. How does the approximation using normal distribution compare to an exact binomial calculation?
3. What happens to the probability if the class size increases to 300?
4. How would you compute the probability of more than 30 students not finishing within an hour?
5. Could we use a Poisson approximation for this problem? Why or why not?

**Tip**: When using normal approximation to binomial, always apply a continuity correction by adding or subtracting 0.5 to the value you are approximating.

This problem explores the probability of 12 or fewer students out of 240 not completing a statistics test in under an hour, using binomial distribution and normal approximation. Learn how to calculate probabilities with statistical methods, including key concepts like normal approximation and the continuity correction.

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