A pharmaceutical company knows that five percent of all users of a certain drug experience a serious side effectA researcher examines a sample of 230 users of the drug . aWhat is the probability of finding between 8 and 12 cases with side effects? Note: Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places

This problem can be solved using the binomial distribution, since we are looking for the probability of a specific number of occurrences (side effects) in a sample of users.

The binomial distribution formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}
\]

where:
- \(n = 230\) is the number of users in the sample,
- \(p = 0.05\) is the probability of a user experiencing side effects,
- \(k\) is the number of users experiencing side effects (in this case, between 8 and 12).

The problem asks for the probability of finding **between 8 and 12 cases** with side effects, so we need to find:

\[
P(8 \leq X \leq 12) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
\]

This is quite tedious to calculate manually, so we can use the binomial cumulative distribution to solve this. However, an approximation can also be made using the normal distribution since \(n\) is large and \(p\) is small, making the binomial distribution approximately normal.

### Step 1: Approximating Binomial with Normal Distribution
The mean and standard deviation of the binomial distribution are:
\[
\mu = np = 230 \times 0.05 = 11.5
\]
\[
\sigma = \sqrt{np(1 - p)} = \sqrt{230 \times 0.05 \times 0.95} \approx 3.297
\]

Now, we can approximate the probability by converting the binomial problem to the normal distribution.

### Step 2: Applying Normal Approximation
Using the continuity correction, we want to find:

\[
P(7.5 \leq X \leq 12.5)
\]

Standardizing this using the z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

For \(X = 7.5\):
\[
Z = \frac{7.5 - 11.5}{3.297} \approx -1.215
\]

For \(X = 12.5\):
\[
Z = \frac{12.5 - 11.5}{3.297} \approx 0.303
\]

### Step 3: Finding the Probabilities
Using standard normal distribution tables or a calculator, we find:

\[
P(Z \leq -1.215) \approx 0.1122
\]
\[
P(Z \leq 0.303) \approx 0.6180
\]

Thus, the probability of finding between 8 and 12 cases with side effects is approximately:

\[
P(7.5 \leq X \leq 12.5) = 0.6180 - 0.1122 = 0.5058
\]

### Final Answer:
The probability of finding between 8 and 12 cases with side effects is approximately **0.5058**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you calculate the binomial probability directly without approximation?
2. What is the exact probability for each value of \(k\) using the binomial formula?
3. How does the continuity correction improve normal approximation for discrete distributions?
4. What is the z-score and how is it used in normal distribution calculations?
5. What factors determine when a binomial distribution can be approximated by a normal distribution?

**Tip:** When using normal approximation to a binomial distribution, don't forget to apply the continuity correction for more accurate results!

A pharmaceutical company knows that five percent of all users of a certain drug experience a serious side effect. A researcher examines a sample of 230 users of the drug. What is the probability of finding between 8 and 12 cases with side effects? Note: Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.

Calculate the probability of finding between 8 and 12 cases of serious side effects in a sample of 230 drug users. This problem uses binomial distribution and normal approximation concepts. Ideal for college-level statistics and probability studies.

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