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%).

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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).