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According to a report, [tex]$73.8\%$[/tex] 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 245 murders by firearm in a random sample of 300 murders? Why?

(a) We would expect [tex]$\square$[/tex] to be committed with a firearm.

(b) Choose the correct answer below.
- A. No, because 245 is less than [tex]$\mu-2 \sigma$[/tex].
- B. Yes, because 245 is between [tex]$\mu-2 \sigma$[/tex] and [tex]$\mu+2 \sigma$[/tex].
- C. No, because 245 is greater than [tex]$\mu+2 \sigma$[/tex].
- D. Yes, because 245 is greater than [tex]$\mu+2 \sigma$[/tex].
- E. No, because 245 is between [tex]$\mu-2 \sigma$[/tex] and [tex]$\mu+2 \sigma$[/tex].



Answer :

GinnyAnswer
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].

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