According to a report, 74.1% of murders are committed with a firearm. (a) If 200 murders are randomly selected, how many would we expect to be committed with a firearm? (b) Would it be unusual to observe 167 murders by firearm in a random sample of 200 murders? Why? (a) We would expect nothing to be committed with a firearm. (b) Choose the correct answer below. A. No, because 167 is less than mu minus 2 sigma. B. Yes, because 167 is between mu minus 2 sigma and mu plus 2 sigma. C. Yes, because 167 is greater than mu plus 2 sigma .D. No, because 167 is between mu minus 2 sigma and mu plus 2 sigma . E. No, because 167 is greater than mu plus 2 sigma.

(a) We would expect 148.2 murders to be committed with a firearm.

(b) Yes, because 167 is greater than μ + 2σ .

Step-by-step explanation:

Let X = number of murders that are committed with a firearm.

The probability that a murder is committed with a firearm is, p = 0.741.

(a)

A random sample of n = 200 murders are selected.

A murder being committed with a firearm is independent o the others.

The random variable X follows a Binomial distribution with parameters n = 200 and p = 0.741.

The expected value of a Binomial random variable is:

E (X) = n × p

Compute the expected number of murder committed with a firearm in the sample of 200 murders as follows:

E (X) = n × p

        = 200 × 0.741

        = 148.2

Thus, the expected number of murder committed with a firearm is 148.2.

(b)

According to the rule of thumb, data values that are more than two standard deviations away from the mean are considered as unusual.

That is, if X is unusual then:

X < μ - 2σ or X > μ + 2σ

The value that is considered unusual here is,

X = 167.

Check whether 167 murders with firearm are unusual or not as follows:

μ ± 2σ = np ± (2 × √np(1-  p))

          = 148.2 ± 6.1955

          = (142.0045, 154.3955)

          ≈ (142, 154)

The value 167 lies outside this range or X > μ + 2σ ⇒ 167 > 154.

Thus, concluding that it would be unusual to observe 167 murders by firearm in a random sample of 200 murders.

Correct option:

Yes, because 167 is greater than μ + 2σ .