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

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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:
  $\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{50 \cdot 0.59 \cdot 0.41} \approx \sqrt{12.095} \approx 3.478$

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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:

$P(26 \leq X \leq 33) = P(Z \leq 1.15) - P(Z \leq -1.15) = 0.8749 - 0.1251 = 0.7498$

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3. Final Answer:

The probability that between 26 and 33 of the victims knew their murderer is approximately:

$\boxed{0.7498}$

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

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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.