To solve this problem, we can use the normal distribution approximation to the binomial distribution. Here's a step-by-step breakdown of how we will approach it:

Step 1: Define the binomial distribution

• The probability $p$ that a murder victim knew their murderer is 63%, or $p = 0.63$.
• The number of cases is $n = 57$, which represents the number of unsolved murders in Detective Williams's case load.
• The mean $\mu$ and standard deviation $\sigma$ of the binomial distribution are given by:

$\mu = np = 57 \times 0.63 = 35.91$ $\sigma = \sqrt{np(1-p)} = \sqrt{57 \times 0.63 \times (1 - 0.63)} = \sqrt{57 \times 0.63 \times 0.37} \approx 3.64$

Now we can use this information to approximate the binomial distribution with a normal distribution $N(\mu = 35.91, \sigma = 3.64)$.

Step 2: Apply continuity correction

Since we are using a normal distribution to approximate a discrete binomial distribution, we apply the continuity correction by adjusting the discrete values by 0.5.

Part (a) - Probability that at least 31 victims knew their murderer

We are asked to find the probability that at least 31 of the victims knew their murderer. This is equivalent to finding:

$P(X \geq 31)$

With continuity correction, this becomes:

$P(X \geq 30.5)$

We convert this to a z-score:

$z = \frac{x - \mu}{\sigma} = \frac{30.5 - 35.91}{3.64} \approx -1.49$

Using standard normal distribution tables or a calculator, the probability associated with $z = -1.49$ is approximately 0.0681. So, the probability that fewer than 31 victims knew their murderer is 0.0681. Therefore, the probability that at least 31 victims knew their murderer is:

$P(X \geq 31) = 1 - 0.0681 = 0.9319$

Thus, the probability is approximately 0.9319 or 93.19%.

Part (b) - Probability that between 31 and 42 victims knew their murderer

We are asked to find the probability that between 31 and 42 victims knew their murderer, i.e.:

$P(31 \leq X \leq 42)$

With continuity correction, this becomes:

$P(30.5 \leq X \leq 42.5)$

We calculate the z-scores for 30.5 and 42.5:

$z_1 = \frac{30.5 - 35.91}{3.64} \approx -1.49$ $z_2 = \frac{42.5 - 35.91}{3.64} \approx 1.81$

Using the standard normal distribution table, the probability for $z_1 = -1.49$ is approximately 0.0681, and for $z_2 = 1.81$, it is approximately 0.9649.

Therefore, the probability that between 31 and 42 victims knew their murderer is:

$P(31 \leq X \leq 42) = 0.9649 - 0.0681 = 0.8968$

Thus, the probability is approximately 0.8968 or 89.68%.

Summary:

• (a) The probability that at least 31 victims knew their murderer is approximately 93.19%.
• (b) The probability that between 31 and 42 victims knew their murderer is approximately 89.68%.

Would you like more details on any step? Here are some related questions to expand on this:

1. How do we derive the formulas for the mean and standard deviation in a binomial distribution?
2. What exactly is the continuity correction, and why do we apply it in normal approximations?
3. How do you find z-scores using a standard normal distribution table?
4. How accurate is the normal approximation to the binomial distribution in this context?
5. Can we solve these types of problems using other methods, like simulation?

Tip: Whenever you approximate a binomial distribution with a normal distribution, remember to use the continuity correction by adding or subtracting 0.5 to the discrete values.