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To help you solve this hypothesis testing problem, let's go through it step-by-step.

### Problem Summary:
1. **Population Parameters**:
   - Population mean age (\(\mu\)) when smokers first start: **13 years**
   - Population standard deviation (\(\sigma\)): **1.9 years**

2. **Sample Data**:
   - Sample mean age (\(\bar{x}\)) of smokers in the current generation: **12 years**
   - Sample size (\(n\)): **35**

3. **Significance Level**:
   - \(\alpha = 0.10\) (10% significance level)

4. **Claim**:
   - To determine if the mean starting age has **changed** from 13 years. Since "changed" implies a two-tailed test, we’ll set up our hypotheses accordingly.

### Step 1: Formulate the Hypotheses
For a two-tailed test:
   - **Null Hypothesis (\(H_0\))**: \(\mu = 13\) years (no change)
   - **Alternative Hypothesis (\(H_1\))**: \(\mu \neq 13\) years (there is a change)

### Step 2: Calculate the Test Statistic (z)
Using the formula for the z-test for a single sample mean:
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
where:
   - \(\bar{x} = 12\)
   - \(\mu = 13\)
   - \(\sigma = 1.9\)
   - \(n = 35\)

Plugging in these values:
\[
z = \frac{12 - 13}{\frac{1.9}{\sqrt{35}}} = \frac{-1}{\frac{1.9}{\sqrt{35}}}
\]

Calculating further:

\[
z \approx \frac{-1}{0.3206} \approx -3.12
\]

### Step 3: Find the p-value
Since this is a two-tailed test, we need to find the p-value for \(z = -3.12\) and double it.

Using the standard normal distribution:
   - The p-value associated with \(z = -3.12\) is approximately \(0.0009\).
   - Since it’s a two-tailed test, we multiply by 2:
     \[
     \text{p-value} = 2 \times 0.0009 = 0.0018
     \]

### Step 4: Decision Rule
- Compare the p-value (0.0018) to the significance level (0.10).
   - Since \(0.0018 < 0.10\), we **reject the null hypothesis**.

### Step 5: Conclusion
The data **supports the claim** that the mean starting age for smokers has changed from 13 years.

### Summary of Answers
1. **Hypotheses**:
   - \(H_0\): \(\mu = 13\) years
   - \(H_1\): \(\mu \neq 13\) years

2. **Test Statistic (z)**: \(-3.12\)
3. **p-value**: \(0.0018\)
4. **Decision**: Reject the null hypothesis
5. **Conclusion**: There is sufficient evidence to support that the mean age when smokers first start is different from 13.

Would you like more details on any part of this process, or do you have any questions?

### Related Questions
1. How would the conclusion change if the significance level was set at 5%?
2. What would the hypotheses look like for a one-tailed test claiming a decrease in mean starting age?
3. How do we interpret the p-value in practical terms for this study?
4. If the sample size was smaller, how might this affect the outcome of the test?
5. How can confidence intervals be used to further validate the claim?

### Tip
Always check if the test is one-tailed or two-tailed based on the wording of the research question, as this impacts the hypotheses and interpretation of results.

The mean age when smokers first start is 13 years old with a population standard deviation of 1.9 years. A researcher thinks that smoking age has significantly changed since the invention of ENDS—electronic nicotine delivery systems. A survey of smokers of this generation was done to see if the mean age has changed. The sample of 35 smokers found that their mean starting age was 12 years old. Do the data support the claim at the 10% significance level? Determine the hypotheses, test statistic, p-value, and conclusion.

Learn to conduct hypothesis testing on whether the mean age when smokers first start has changed from 13 years using a 10% significance level. This problem covers setting up null and alternative hypotheses, calculating the z-test statistic, and determining the p-value for decision-making in hypothesis testing.

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