Let's break down the problem and solve it step by step using a normal approximation to the binomial distribution.

Given:

• Probability that a murder victim knew their murderer $p = 0.59$
• Number of unsolved cases $n = 49$
• We are interested in finding probabilities using a normal approximation to the binomial distribution.

To apply the normal approximation, we first calculate the mean and standard deviation of the binomial distribution.

1. Mean and Standard Deviation

For a binomial distribution, the mean $\mu$ and the standard deviation $\sigma$ are given by:

• $\mu = n \times p$
• $\sigma = \sqrt{n \times p \times (1 - p)}$

Mean:

$\mu = 49 \times 0.59 = 28.91$

Standard Deviation:

$\sigma = \sqrt{49 \times 0.59 \times 0.41} = \sqrt{11.8299} \approx 3.44$

Now, we can proceed with the specific questions using the normal distribution.

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a. Probability that at least 25 victims knew their murderer.

We need to find $P(X \geq 25)$. Using the normal approximation, we apply a continuity correction by subtracting 0.5 from 25. This gives us $P(X \geq 24.5)$.

First, we calculate the corresponding z-score: $z = \frac{24.5 - \mu}{\sigma} = \frac{24.5 - 28.91}{3.44} \approx \frac{-4.41}{3.44} \approx -1.28$

Using the standard normal distribution table or a calculator, the probability corresponding to $z = -1.28$ is approximately 0.1003. This is the probability that $X < 25$, so the probability that $X \geq 25$ is:

$P(X \geq 25) = 1 - 0.1003 = 0.8997$

Thus, the probability that at least 25 of the victims knew their murderer is approximately 0.8997 (or 89.97%).

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b. Probability that between 25 and 33 of the victims knew their murderer.

We need to find $P(25 \leq X \leq 33)$. Using continuity correction, this becomes $P(24.5 \leq X \leq 33.5)$.

For $X = 24.5$:

The z-score was calculated earlier as $z = -1.28$.

For $X = 33.5$:

$z = \frac{33.5 - 28.91}{3.44} = \frac{4.59}{3.44} \approx 1.33$

Using the standard normal table, the probability corresponding to $z = 1.33$ is approximately 0.9082.

Now, we find the probability that $X$ is between 24.5 and 33.5: $P(25 \leq X \leq 33) = P(24.5 \leq X \leq 33.5) = 0.9082 - 0.1003 = 0.8079$

Thus, the probability that between 25 and 33 of the victims knew their murderer is approximately 0.8079 (or 80.79%).

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Summary:

• a. The probability that at least 25 victims knew their murderer is 0.8997.
• b. The probability that between 25 and 33 victims knew their murderer is 0.8079.

Would you like more details or have any other questions?

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Here are 5 related questions you might explore:

1. How would the probabilities change if the percentage of victims who knew their murderer was higher, say 70%?
2. What is the probability that fewer than 20 victims knew their murderer?
3. How does using a normal approximation compare to using an exact binomial probability?
4. How would the calculations change if there were 100 unsolved cases instead of 49?
5. What is the probability that more than 40 victims knew their murderer?

Tip: When using normal approximations to binomial distributions, always remember to apply the continuity correction for better accuracy.