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To find the 90% confidence interval for the mean starting age of smokers, we can use the formula for the confidence interval when the population standard deviation is known:

\[
\text{CI} = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}
\]

where:
- \(\bar{x} = 13.4\) (sample mean),
- \(\sigma = 1.3\) (population standard deviation),
- \(n = 21\) (sample size),
- \(Z\) is the Z-score corresponding to a 90% confidence level (for a 90% confidence interval, \(Z \approx 1.645\)).

Let's plug in the values.

### Step 1: Calculate the standard error
\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{1.3}{\sqrt{21}} \approx 0.2839
\]

### Step 2: Calculate the margin of error
\[
\text{Margin of Error} = Z \cdot \text{Standard Error} = 1.645 \times 0.2839 \approx 0.4666
\]

### Step 3: Construct the confidence interval
\[
\text{CI} = 13.4 \pm 0.4666
\]

This gives:
\[
\text{Lower Limit} = 13.4 - 0.4666 \approx 12.933
\]
\[
\text{Upper Limit} = 13.4 + 0.4666 \approx 13.867
\]

### Answer:
The 90% confidence interval for the mean starting age is approximately:
\[
(12.933, 13.867)
\]

Do you want more details or have any questions?

### Related Questions
1. How is the Z-score determined for different confidence levels?
2. What if the sample size were larger? How would that affect the interval?
3. How does the population standard deviation impact the confidence interval width?
4. Why is a Z-distribution used instead of a T-distribution in this case?
5. How would we calculate this if the population standard deviation was unknown?

**Tip:** Remember that larger sample sizes generally yield narrower confidence intervals, increasing the precision of the estimate.

The age when smokers first start from previous studies is normally distributed with a mean of 13 years old with a population standard deviation of 1.3 years old. A survey of smokers of this generation was done to estimate if the mean age has changed. The sample of 21 smokers found that their mean starting age was 13.4 years old. Find the 90% confidence interval of the mean.

Learn how to find the 90% confidence interval for the mean starting age of smokers. This solution involves using the normal distribution, population standard deviation, and sample size to calculate the interval. Includes step-by-step instructions for understanding margin of error and standard error.

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