Question

Suppose that about 55% of the people who are murdered actually knew the person who committed the murder. Suppose that a detective file in Boston has 51 current unsolved murders. What is the probability that fewer than 21 victims did not know their murderer?

Answer 1

Answer:

24.51% probability that fewer than 21 victims did not know their murderer

Step-by-step explanation:

I am going to approximate the binomial probability distribution to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$n = 51, p = 0.55$

So

$\mu = E(X) = np = 51*0.55 = 28.05$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{51*0.55*0.45} = 3.55$

What is the probability that fewer than 21 victims did not know their murderer?

That is, more than 30 knew the murdered.

$P(X \geq 30 + 0.5) = P(X \geq 30.5)$, which is 1 subtracted by the pvalue of Z when X = 30.5. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{30.5 - 28.05}{3.55}$

$Z = 0.69$

$Z = 0.69$ has a pvalue of 0.7549

1 - 0.7549 = 0.2451

24.51% probability that fewer than 21 victims did not know their murderer