Let . Use the Normal approximation (with the continuity correction) to calculate P(X > 64). Give your answer to 4 decimal places.

1. Break Down the Problem

We need to calculate $P(X > 64)$ using the Normal approximation with continuity correction for a binomial distribution.

2. Relevant Concepts

1. Binomial Distribution Parameters:

   • Let $n$ be the number of trials.
   • Let $p$ be the probability of success on each trial.
   • Therefore, $X$ follows a binomial distribution $X \sim \text{Binomial}(n, p)$.

2. Normal Approximation:

   • When $n$ is large, the binomial distribution can be approximated by a normal distribution: $X \sim N(\mu, \sigma^2)$ where
   • $\mu = np$ (mean),
   • $\sigma^2 = np(1-p)$ (variance).

3. Continuity Correction:

   • To approximate $P(X > 64)$, we use the continuity correction: $P(X > 64) \approx P(X \geq 65) \approx P\left( Z > \frac{64.5 - \mu}{\sigma} \right)$ where $Z$ follows the standard normal distribution $Z \sim N(0, 1)$.

3. Analysis and Detail

• Calculate $\mu$ and $\sigma$ with the given $n$ and $p$:
  • Suppose $n$ and $p$ are given (add values here if provided).

1. Mean Calculation: $\mu = np$

2. Variance Calculation: $\sigma^2 = np(1-p) \quad \Rightarrow \quad \sigma = \sqrt{np(1-p)}$

3. Calculate Z-score: $Z = \frac{64.5 - \mu}{\sigma}$

4. Verify and Summarize

• After obtaining $Z$, look up the corresponding value in the standard normal distribution table or use a calculator to find $P(Z > z)$.

Final Answer

• Provide the probability value to four decimal places after calculating from the Z-score.

(Note: Please provide the values of $n$ and $p$ to complete the calculations.)