About 59% of people who are murdered knew the person who committed the murder.

Suppose that Detective Smith's case load currently has 50 unsolved murders. Use a normal distribution approximation to the binomial distribution to find the following probabilities:

a. What is the probability that at least 26 of the victims among Detective Smith's case files knew their murderer?

0.875

b. What is the probability that between 26 and 33 of the victims knew their murderer?

To solve this problem using the **normal approximation to the binomial distribution**, let’s walk through it step-by-step.

---

### 1. Define the parameters:

- \( n = 50 \) (number of cases)
- \( p = 0.59 \) (probability that the victim knew their murderer)
- \( q = 1 - p = 0.41 \) (probability the victim did not know the murderer)

We need to calculate the probabilities using a **normal approximation** to the binomial distribution. 

For normal approximation:
- **Mean**:  
  \[ \mu = n \cdot p = 50 \cdot 0.59 = 29.5 \]
  
- **Standard deviation**:  
  \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{50 \cdot 0.59 \cdot 0.41} \approx \sqrt{12.095} \approx 3.478 \]

---

### 2. Part (b): What is the probability that between **26 and 33** of the victims knew their murderer?

This is a **range probability** for \( 26 \leq X \leq 33 \).

#### Apply continuity correction:
Since we are using the normal approximation, we adjust the boundaries slightly:
- Lower boundary: \( 26 - 0.5 = 25.5 \)
- Upper boundary: \( 33 + 0.5 = 33.5 \)

We now convert these values to **z-scores**.

#### Z-scores for the boundaries:
\[
z_1 = \frac{25.5 - \mu}{\sigma} = \frac{25.5 - 29.5}{3.478} \approx \frac{-4}{3.478} \approx -1.15
\]
\[
z_2 = \frac{33.5 - \mu}{\sigma} = \frac{33.5 - 29.5}{3.478} \approx \frac{4}{3.478} \approx 1.15
\]

#### Use z-table or normal CDF to find probabilities:
\[
P(Z \leq -1.15) \approx 0.1251
\]
\[
P(Z \leq 1.15) \approx 0.8749
\]

#### Compute the final probability:
\[
P(26 \leq X \leq 33) = P(Z \leq 1.15) - P(Z \leq -1.15) = 0.8749 - 0.1251 = 0.7498
\]

---

### 3. Final Answer:
The probability that between **26 and 33** of the victims knew their murderer is approximately:

\[
\boxed{0.7498}
\]

---

### Do you have any follow-up questions or need further clarifications?

---

Here are **5 related questions** to expand your knowledge:  
1. How does the continuity correction improve the accuracy of the normal approximation?  
2. When is it appropriate to use a normal approximation for binomial problems?  
3. How do you calculate binomial probabilities exactly without approximation?  
4. What are the limitations of the normal approximation to the binomial distribution?  
5. How would the results change if the probability \( p \) was closer to 0.5 or 1?

**Tip**: The normal approximation becomes more accurate when \( n \) is large, and \( p \) is not too close to 0 or 1.

About 59% of people who are murdered knew the person who committed the murder.

Suppose that Detective Smith's case load currently has 50 unsolved murders. Use a normal distribution approximation to the binomial distribution to find the following probabilities:

a. What is the probability that at least 26 of the victims among Detective Smith's case files knew their murderer?

b. What is the probability that between 26 and 33 of the victims knew their murderer?

Learn how to apply the normal approximation to the binomial distribution for probability questions related to unsolved murder cases. This example calculates the probability that at least 26, and between 26 and 33, victims knew their murderer. The solution uses key statistical concepts like binomial distribution, continuity correction, and z-scores.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation