To solve this problem using the Central Limit Theorem, we'll follow these steps:

1. Identify the parameters of the binomial distribution.
2. Use the Central Limit Theorem to approximate the binomial distribution with a normal distribution.
3. Calculate the probability that between 28 and 33 people knew their murderer.

Step 1: Define the Parameters

We have a binomial distribution because each cold case has two possible outcomes: the victim knew the murderer or they didn't. The parameters are:

• Number of trials ($n$) = 50 (total cold cases)
• Probability of success ($p$) = 0.60 (probability that the victim knew their murderer)

Since we’re using the Central Limit Theorem, we approximate this binomial distribution with a normal distribution.

Step 2: Apply the Central Limit Theorem

The Central Limit Theorem tells us that for a large enough $n$, a binomial distribution can be approximated by a normal distribution with:

• Mean $\mu = np$
• Standard deviation $\sigma = \sqrt{np(1 - p)}$

Calculate $\mu$ and $\sigma$:

$\mu = 50 \times 0.60 = 30$ $\sigma = \sqrt{50 \times 0.60 \times 0.40} = \sqrt{12} \approx 3.46$

So, we approximate our binomial distribution with a normal distribution $N(30, 3.46)$.

Step 3: Calculate the Probability

We want to find the probability that between 28 and 33 of the victims knew their murderer. This means calculating:

$P(28 \leq X \leq 33)$

To use the normal distribution, we perform a continuity correction by adjusting our bounds to 27.5 and 33.5.

Convert to Standard Normal Distribution (Z-scores)

The Z-scores for 27.5 and 33.5 are:

$Z = \frac{27.5 - 30}{3.46} \approx -0.72$ $Z = \frac{33.5 - 30}{3.46} \approx 1.01$

Using standard normal distribution tables or a calculator, we find the probabilities for these Z-scores:

• For $Z = -0.72$, $P(Z < -0.72) \approx 0.2358$
• For $Z = 1.01$, $P(Z < 1.01) \approx 0.8438$

Now, calculate the probability that $Z$ is between $-0.72$ and $1.01$:

$P(-0.72 \leq Z \leq 1.01) = 0.8438 - 0.2358 = 0.608$

Answer

The probability that between 28 and 33 of the 50 victims knew their murderer is approximately 0.61 (or 61%).

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Related Questions

1. How would the probability change if the probability of knowing the murderer were 70% instead of 60%?
2. What would the probability be for a larger sample, say 100 cold cases?
3. How does the Central Limit Theorem help in approximating probabilities in binomial distributions?
4. How does changing the sample size affect the standard deviation in this problem?
5. What if we wanted to calculate the exact binomial probability without using the Central Limit Theorem?

Tip

When approximating a binomial distribution using the Central Limit Theorem, always remember to apply the continuity correction to improve accuracy.