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.