10) Terrie Moffitt, the author of the dual taxonomy theory, proposed that there is a group of
offenders known as the life course persistent offenders who commit both frequent and serious
offenses throughout their lives. Suppose you have a sample of 25 young males who have been in
juvenile institutions at least twice. The mean age of their first arrest was 12 and the standard
deviation is 1.9. Construct a 95 percent confidence interval around this point estimate.
a) the intervals are 11.84 and 12.16
b) the intervals are 11.20 and 12.80
c) the intervals are 10.78 and 14.22
d) none of the above

To construct a 95% confidence interval for the mean age of the first arrest, we'll use the following formula:

\[
\text{Confidence Interval} = \bar{x} \pm z \times \frac{s}{\sqrt{n}}
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(z\) is the z-value corresponding to the desired confidence level.
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.

### Step 1: Identify the values
- Sample mean, \(\bar{x}\) = 12
- Standard deviation, \(s\) = 1.9
- Sample size, \(n\) = 25
- For a 95% confidence interval, the z-value (\(z\)) is approximately 1.96.

### Step 2: Calculate the standard error
\[
\text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{1.9}{\sqrt{25}} = \frac{1.9}{5} = 0.38
\]

### Step 3: Calculate the margin of error
\[
\text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 0.38 = 0.7448
\]

### Step 4: Calculate the confidence interval
\[
\text{Confidence Interval} = \bar{x} \pm \text{Margin of Error} = 12 \pm 0.7448
\]
\[
\text{Lower bound} = 12 - 0.7448 = 11.2552
\]
\[
\text{Upper bound} = 12 + 0.7448 = 12.7448
\]

So, the 95% confidence interval is approximately \(11.26\) to \(12.74\), which is not exactly matching any of the given choices. However, the closest choice would be **b) the intervals are 11.20 and 12.80**.

### Final Answer:
**b) the intervals are 11.20 and 12.80**

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you interpret a confidence interval in the context of this problem?
2. What happens to the confidence interval if we increase the sample size?
3. How would the confidence interval change if the standard deviation were larger?
4. What is the significance of the z-value in constructing a confidence interval?
5. How would you calculate a confidence interval if the sample size were small (e.g., \(n < 30\))?

**Tip:** The width of the confidence interval depends on both the sample size and the standard deviation; increasing the sample size generally leads to a narrower interval, making the estimate more precise.

Learn how to construct a 95% confidence interval for the mean age of first arrest using sample mean, standard deviation, and sample size. This problem involves applying statistical concepts to analyze criminal behavior among young males in juvenile institutions.

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