To find $P(X > 32)$ using the normal approximation, we follow these steps:

Step 1: Compute the mean and standard deviation of the binomial distribution

The mean $\mu$ and standard deviation $\sigma$ of a binomial distribution are given by:

$\mu = n \cdot p$ $\sigma = \sqrt{n \cdot p \cdot (1 - p)}$

Given:

• $n = 85$
• $p = 0.48$

Now, calculate the mean and standard deviation:

$\mu = 85 \cdot 0.48 = 40.8$ $\sigma = \sqrt{85 \cdot 0.48 \cdot (1 - 0.48)} = \sqrt{85 \cdot 0.48 \cdot 0.52} \approx 4.6247$

Step 2: Use the continuity correction

Since we are using the normal approximation, we apply a continuity correction. For $P(X > 32)$, we calculate $P(X > 32.5)$.

Step 3: Convert to the standard normal variable

We convert $X = 32.5$ to a $Z$-score using the formula:

$Z = \frac{X - \mu}{\sigma}$

Substitute the values:

$Z = \frac{32.5 - 40.8}{4.6247} \approx \frac{-8.3}{4.6247} \approx -1.7942$

Step 4: Find the probability from the cumulative normal distribution table

Using the cumulative normal distribution table, find the probability corresponding to $Z = -1.7942$.

For $Z \approx -1.79$, the cumulative probability is approximately 0.0367.

Step 5: Calculate the required probability

Since we are looking for $P(X > 32)$, we need to subtract this cumulative probability from 1:

$P(X > 32) = 1 - 0.0367 = 0.9633$

Final Answer:

$P(X > 32) \approx 0.9633$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the continuity correction affect normal approximation?
2. How would the probability change if $p$ were higher, say 0.6?
3. Why do we use the standard normal distribution in binomial probability problems?
4. Can the normal approximation be used for small sample sizes?
5. How does the cumulative normal distribution table help in these problems?

Tip: Always check if $np$ and $n(1 - p)$ are both greater than 5 when using the normal approximation for binomial probabilities.