Sure, let's go through the details for each part of the question step-by-step.
### (a) Expected Number of Murders Committed with a Firearm
Given that 73.8% of murders are committed with a firearm, we want to calculate the expected number of murders by firearm out of a total of 300 murders.
1. Percentage in Decimal Form: Convert 73.8% into a decimal.
[tex]\[
0.738
\][/tex]
2. Expected Number of Firearm Murders: Multiply the total number of murders by the percentage in decimal form:
[tex]\[
\text{Expected Firearm Murders} = 300 \times 0.738 = 221.4
\][/tex]
So, we would expect [tex]\(\boxed{221.4}\)[/tex] murders to be committed with a firearm.
### (b) Determining if 245 Murders by Firearm is Unusual
We need to determine whether observing 245 murders by firearm in a sample of 300 is unusual based on the distribution of murders committed with firearms.
1. Mean ([tex]\(\mu\)[/tex]): We've already calculated this in part (a):
[tex]\[
\mu = 221.4
\][/tex]
2. Standard Deviation ([tex]\(\sigma\)[/tex]):
The standard deviation for the number of murders by firearm can be computed using the binomial standard deviation formula:
[tex]\[
\sigma = \sqrt{n \times p \times (1 - p)}
\][/tex]
Where [tex]\(n = 300\)[/tex] is the total number of murders and [tex]\(p = 0.738\)[/tex] is the probability:
[tex]\[
\sigma = \sqrt{300 \times 0.738 \times (1 - 0.738)} \approx 7.635
\][/tex]
3. Calculate Z-score for 245 Murders:
The z-score formula is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
For [tex]\(x = 245\)[/tex]:
[tex]\[
z = \frac{245 - 221.4}{7.635} \approx 3.099
\][/tex]
4. Determine if 245 is Unusual:
An observation is generally considered unusual if its z-score is outside the range [tex]\([-2, 2]\)[/tex], which means it is significantly far from the mean.
Since [tex]\(z \approx 3.099\)[/tex] is greater than 2, 245 murders by firearm is outside this range and thus considered unusual.
### Conclusion for (b)
245 murders by firearm would be considered unusual because it is greater than [tex]\(\mu + 2\sigma\)[/tex].
So the correct answer for part (b) is:
- D. Yes, because 245 is greater than [tex]\(\mu + 2 \sigma\)[/tex].
