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?



To solve this problem using the normal approximation to the binomial distribution, we will follow these steps:

### Step 1: Problem Restatement and Binomial Setup

The problem mentions that:
- The probability (\( p \)) that a person who was murdered knew the murderer is 59%, or \( p = 0.59 \).
- The number of unsolved murders in Detective Smith's case load is \( n = 50 \).

This is a binomial distribution problem with:
- \( n = 50 \)
- \( p = 0.59 \)

However, instead of solving it directly using the binomial distribution, we'll approximate it using the normal distribution.

### Step 2: Normal Approximation to the Binomial Distribution

For normal approximation, the mean \( \mu \) and standard deviation \( \sigma \) of the binomial distribution are given by:

\[
\mu = n \cdot p = 50 \cdot 0.59 = 29.5
\]

\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{50 \cdot 0.59 \cdot (1 - 0.59)} \approx \sqrt{50 \cdot 0.59 \cdot 0.41} \approx \sqrt{12.095} \approx 3.478
\]

Thus, we will use \( \mu = 29.5 \) and \( \sigma = 3.478 \) for the normal distribution.

### Step 3: Continuity Correction

Since we are approximating a discrete binomial distribution with a continuous normal distribution, we apply the continuity correction.

### Part (a): Probability that at least 26 of the victims knew their murderer

We need to calculate \( P(X \geq 26) \), which translates to:

\[
P(X \geq 26) \approx P(X > 25.5)
\]

We now convert this to a standard normal \( Z \)-score using the formula:

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

For \( X = 25.5 \):

\[
Z = \frac{25.5 - 29.5}{3.478} \approx \frac{-4}{3.478} \approx -1.15
\]

Using standard normal tables (or a calculator), we find the probability corresponding to \( Z = -1.15 \). The cumulative probability for \( Z = -1.15 \) is approximately 0.1251. Thus,

\[
P(X \geq 26) = 1 - 0.1251 = 0.8749
\]

So, the probability that at least 26 victims knew their murderer is approximately **0.8749** (or **87.49%**).

### Part (b): Probability that between 26 and 33 of the victims knew their murderer

Here, we need to calculate \( P(26 \leq X \leq 33) \), which translates to:

\[
P(26 \leq X \leq 33) \approx P(25.5 \leq X \leq 33.5)
\]

We will compute the two Z-scores for \( X = 25.5 \) and \( X = 33.5 \).

- For \( X = 25.5 \):

\[
Z = \frac{25.5 - 29.5}{3.478} \approx -1.15
\]

- For \( X = 33.5 \):

\[
Z = \frac{33.5 - 29.5}{3.478} \approx 1.15
\]

From standard normal tables:
- The cumulative probability for \( Z = -1.15 \) is approximately 0.1251.
- The cumulative probability for \( Z = 1.15 \) is approximately 0.8749.

Thus, the probability that between 26 and 33 victims knew their murderer is:

\[
P(26 \leq X \leq 33) = 0.8749 - 0.1251 = 0.7498
\]

So, the probability that between 26 and 33 victims knew their murderer is approximately **0.7498** (or **74.98%**).

---

### Summary of Results:
- (a) The probability that at least 26 victims knew their murderer: **0.8749** (87.49%).
- (b) The probability that between 26 and 33 victims knew their murderer: **0.7498** (74.98%).

Do you want further details or explanations on any of the steps? Here are some related questions to explore this topic:

1. How do we compute binomial probabilities without normal approximation?
2. What is the significance of continuity correction in normal approximation?
3. How does the shape of a binomial distribution affect the normal approximation accuracy?
4. What are other real-world applications of the normal approximation to the binomial distribution?
5. How would the probabilities change if \( n \) were larger or smaller?

**Tip:** Always remember to apply the continuity correction when approximating discrete distributions (like the binomial) with continuous ones (like the normal).

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?

Explore the use of the normal approximation to the binomial distribution in solving a real-world problem involving Detective Smith's case load. Learn how to calculate the probability that at least 26, and between 26 and 33, victims knew their murderer using concepts like mean, standard deviation, and continuity correction. Suitable for students in Grades 11-12.

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