To verify whether a Normal model can be used to approximate the distribution of the number of good first serves, the question typically refers to the Binomial to Normal approximation. This approximation is valid if certain conditions are met. Here's how to analyze the given options:

Key Concepts:

1. Binomial Distribution: $X \sim B(n, p)$, where $n$ is the number of trials and $p$ is the probability of success.
2. Conditions for Normal Approximation:
   • $np \geq 10$ and $nq \geq 10$, where $q = 1 - p$. This ensures the distribution is sufficiently symmetric and unimodal for approximation.

Now, let's evaluate the options:

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Option A:

The Normal model is an appropriate approximation because the sample size is sufficiently large.

• This is incomplete, as the sample size alone is not enough; the product of $n$ and $p$, as well as $n$ and $q$, must be considered.

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Option B:

The Normal model is an appropriate approximation because the mean is at least [enter your response here] and the standard deviation is at least [enter your response here].

• This is partially true: the mean and standard deviation play a role, but the condition $np \geq 10$ and $nq \geq 10$ is more relevant.
• The mean of a Binomial distribution is $\mu = np$ and the standard deviation is $\sigma = \sqrt{npq}$.

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Option C:

The Normal model is an appropriate approximation because $np \geq 10$ and $nq \geq 10$.

• This is the correct condition. You must calculate $np$ and $nq$ to confirm they satisfy this requirement.
  $np$ represents the expected number of successes, and $nq$ the expected number of failures. If both are $\geq 10$, then the approximation is valid.

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Option D:

The Normal model is an appropriate approximation because the probability is sufficiently large.

• This is incorrect. While the probability influences the distribution, the criteria depend on $np$ and $nq$, not just $p$ alone.

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Final Steps:

1. Calculate $np$: Determine $np$ by multiplying the total number of trials ($n$) by the probability of success ($p$).
2. Calculate $nq$: Use $nq = n(1-p)$.
3. Verify: Check if both $np$ and $nq$ are $\geq 10$.

If you provide $n$ and $p$, I can help with the exact calculation. Based on this analysis, Option C is most likely the correct choice, as it specifies the necessary conditions.

Would you like me to assist with the calculations? Let me know if you have further questions!

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Related Questions:

1. What is the formula for the mean and standard deviation of a Binomial distribution?
2. Why is $np \geq 10$ and $nq \geq 10$ important for approximation?
3. How does the shape of a Binomial distribution change with $n$ and $p$?
4. When should a continuity correction be applied in Normal approximation?
5. Can we use the Normal approximation for small values of $n$?

Tip:

Always verify both $np$ and $nq$ to ensure the conditions for Normal approximation are satisfied.