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

This problem is essentially a probability problem, where the probability of a serious side effect is known to be 5% (or \( 0.05 \)), and a sample of 230 users is being examined.

Let's break down the information provided:

- **Probability of a serious side effect (p)** = 5% = 0.05
- **Sample size (n)** = 230

### Key Questions to Address:
1. **Expected Number of Users with Side Effects:**
   The expected number of users in the sample who experience a serious side effect can be found using the formula for expectation in a binomial distribution:
   \[
   E(X) = n \times p
   \]
   Substituting the values:
   \[
   E(X) = 230 \times 0.05 = 11.5
   \]
   So, the expected number of users experiencing a serious side effect is **11.5**.

2. **Variance and Standard Deviation:**
   The variance of a binomial distribution is given by:
   \[
   \text{Var}(X) = n \times p \times (1 - p)
   \]
   Substituting the values:
   \[
   \text{Var}(X) = 230 \times 0.05 \times 0.95 = 10.925
   \]
   The standard deviation is the square root of the variance:
   \[
   \text{Std Dev}(X) = \sqrt{10.925} \approx 3.305
   \]

3. **Probability of a Specific Number of Users with Side Effects:**
   If you want to calculate the probability of a specific number of users experiencing a serious side effect, you would use the **binomial probability formula**:
   \[
   P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
   \]
   where \( k \) is the number of users with side effects.

   If you have a specific \( k \) in mind, we can calculate it.

### Questions for Further Exploration:
1. What is the probability that exactly 10 users experience a serious side effect?
2. What is the probability that fewer than 8 users experience a serious side effect?
3. What is the probability that more than 15 users experience a serious side effect?
4. What is the probability that between 8 and 12 users experience a serious side effect?
5. How does the expected number of users with side effects change if the probability of side effects increases to 8%?

### Tip:
For large sample sizes and small probabilities, the binomial distribution can often be approximated by a **normal distribution** for easier calculations.

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.

Explore the probability of serious side effects in a sample of 230 drug users, with an expected probability of 5%. This problem involves key probability concepts such as expected value, variance, and standard deviation, and uses the binomial distribution formula. Suitable for Grades 11-12 or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation