Question

According to a report, 60.1% of murders are committed with a firearm.

(a) If 300 murders are randomly selected, how many would we expect to be committed with a firearm?
(b) Would it be unusual to observe 211 murders by firearm in a random sample of 300 murders? Why?

Answer 1

The expected number of murders committed with a firearm in a random sample of 300 murders is 180.3. It would be considered unusual to observe 211 murders by firearm in a random sample of 300 murders.

Step-by-step explanation:

(a) Expected number of murders committed with a firearm:

To find the expected number of murders committed with a firearm, we multiply the percentage of murders committed with a firearm by the total number of murders:

Expected number of murders with a firearm = 60.1% × 300 = 180.3

(b) Unusual to observe 211 murders by firearm in a random sample of 300 murders:

To determine whether it is unusual to observe 211 murders by firearm in a random sample of 300 murders, we need to use a statistical test. We can calculate the standard deviation of the expected number of murders committed with a firearm using the formula:

Standard deviation = sqrt(N × P × (1-P))

Where N is the sample size and P is the proportion of murders committed with a firearm.

Using the given data, we can calculate the standard deviation:

Standard deviation = sqrt(300 × 0.601 × (1-0.601)) = 8.062

We can then calculate the z-score for 211 murders using the formula:

z = (observed value - expected value) / standard deviation

z = (211 - 180.3) / 8.062 = 3.791

The z-score tells us how many standard deviations away from the mean the observed value is. If the z-score is less than -1.96 or greater than 1.96, the result is considered statistically significant at a significance level of 0.05. In this case, the z-score of 3.791 is greater than 1.96, so it would be considered unusual to observe 211 murders by firearm in a random sample of 300 murders.