The sample size​ n, probability of success​ p, and probability of failure q are given for a binomial experiment. Decide whether you can use the normal distribution to approximate the random variable x.
nequals19
pequals0.24
qequals0.76

To determine if the normal distribution can approximate a binomial distribution for this problem, we need to verify if the sample size \( n \) and the probabilities \( p \) and \( q \) meet the requirements for a normal approximation. The rule of thumb is that both of the following conditions should be satisfied:

1. \( n \cdot p \geq 5 \)
2. \( n \cdot q \geq 5 \)

Given:
- \( n = 19 \)
- \( p = 0.24 \)
- \( q = 0.76 \)

Let's calculate each:

1. **Check \( n \cdot p \):**
   \[
   n \cdot p = 19 \cdot 0.24 = 4.56
   \]

2. **Check \( n \cdot q \):**
   \[
   n \cdot q = 19 \cdot 0.76 = 14.44
   \]

Since \( n \cdot p = 4.56 \), which is less than 5, the first condition is not met. Therefore, we **cannot** use the normal distribution to approximate the binomial distribution for this experiment.

Would you like a more detailed breakdown or have any further questions?

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**Here are 5 related questions to explore:**
1. How can we increase \( n \) to make normal approximation valid in this case?
2. Why is the rule \( n \cdot p \geq 5 \) and \( n \cdot q \geq 5 \) important for approximation?
3. How does the shape of a binomial distribution change as \( n \) increases?
4. When is it more appropriate to use the Poisson approximation instead of the normal approximation?
5. How does the skewness of a binomial distribution relate to \( p \) and \( q \)?

**Tip:** For small sample sizes or extreme probabilities, consider using the exact binomial probability instead of approximations.

The sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Decide whether you can use the normal distribution to approximate the random variable x. n = 19, p = 0.24, q = 0.76

Determine if the normal distribution can approximate a binomial distribution for a sample size of n = 19 with probabilities p = 0.24 and q = 0.76. Check the conditions n * p ≥ 5 and n * q ≥ 5 to decide the validity of the approximation. Suitable for students in Grades 10-12.

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