To analyze whether the data support the claim that the mean starting age of smokers is at least 19 years, we can conduct a one-sample $t$-test. Here’s how to go through the steps:

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): $\mu \geq 19$
• Alternative Hypothesis ($H_a$): $\mu < 19$

Step 2: Identify the Distribution

Since the sample size is less than 30 and the population standard deviation is unknown, we will use the Student's $t$-distribution.

Step 3: Sample Information

• Sample mean ($\bar{X}$): 18.1
• Sample standard deviation (s): 1.3
• Sample size (n): 37

Step 4: Calculate the Test Statistic

The test statistic for a one-sample $t$-test is calculated as follows:

$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$

Where:

• $\mu_0$ is the hypothesized population mean (19 in this case).

Plugging in the values:

$t = \frac{18.1 - 19}{1.3 / \sqrt{37}} = \frac{-0.9}{1.3 / 6.0828} \approx \frac{-0.9}{0.2133} \approx -4.22$

Step 5: Determine the Critical Value and Decision Rule

For a one-tailed test at the 5% significance level ($\alpha = 0.05$) with $n - 1 = 36$ degrees of freedom, we can look up the critical value in the $t$-distribution table.

The critical value for $t_{0.05, 36}$ is approximately $-1.688$.

Step 6: Compare Test Statistic to Critical Value

• Calculated $t$: $-4.22$
• Critical value: $-1.688$

Since $-4.22 < -1.688$, we reject the null hypothesis.

Conclusion

The data provide sufficient evidence to support the claim that the mean starting age of smokers is less than 19 years at the 5% significance level.

Summary of Results

• Test Statistic $t \approx -4.22$
• Distribution: $X \sim t_{36}$

Would you like further details or have any questions? Here are some follow-up questions to consider:

1. How would the result change if the sample mean were higher?
2. What are the implications of rejecting the null hypothesis in this context?
3. How would you interpret the p-value for this test?
4. What assumptions must hold true for the $t$-test to be valid?
5. How might the results differ if the standard deviation were not constant?

Tip: Always ensure that your sample size is adequate to draw reliable conclusions from statistical tests.