To analyze this problem, we will perform a hypothesis test. Here’s how it is structured:

Step 1: Define Hypotheses

• Null Hypothesis ($H_0$): The mean number of cigarettes smoked per day is 31 ($\mu = 31$).
• Alternative Hypothesis ($H_a$): The mean number of cigarettes smoked per day is less than 31 ($\mu < 31$).

This is a one-tailed test because the claim is that the mean is less than 31.

Step 2: Gather Information

• Population mean ($\mu_0$) = 31
• Sample mean ($\bar{x}$) = 28
• Sample size ($n$) = 10
• Sample standard deviation ($s$) = 2.7
• Significance level ($\alpha$) = 0.05

Step 3: Test Statistic

The test statistic for a t-test is calculated as:

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

Substitute the given values:

$t = \frac{28 - 31}{\frac{2.7}{\sqrt{10}}}$

$t = \frac{-3}{\frac{2.7}{3.1623}} = \frac{-3}{0.854} \approx -3.51$

Step 4: Degrees of Freedom and Critical Value

Degrees of freedom ($df$) = $n - 1 = 10 - 1 = 9$.

Using a t-table or statistical software, find the critical t-value for $df = 9$ at $\alpha = 0.05$ (one-tailed):

$t_{critical} = -1.833$

Step 5: Decision Rule

• If $t < t_{critical}$, reject $H_0$.
• If $t \geq t_{critical}$, fail to reject $H_0$.

Step 6: Conclusion

Here, $t = -3.51$, which is less than $-1.833$. Therefore, we reject $H_0$.

Final Answer

There is sufficient evidence at the 0.05 significance level to support the smoker's claim that the average number of cigarettes smoked per day is less than 31.

Would you like more details about this process, or can I clarify anything?

Here are 5 questions that expand on this concept:

1. How would the conclusion change if the sample size increased to 50 while keeping other parameters the same?
2. What role does the significance level ($\alpha$) play in hypothesis testing?
3. How do t-tests differ from z-tests, and when is each appropriate?
4. What assumptions are necessary for performing this t-test?
5. How would the conclusion change if the sample mean was closer to 31, such as 30?

Tip: Always check whether the sample size is large enough for normal approximation or if a t-test is appropriate.