Question 1200425
Certainly, let's perform a hypothesis test to determine if the population mean pain relief time of the new analgesic drug is actually 3 hours.

**1. Set up Hypotheses**

* **Null Hypothesis (H0):** 
    * μ = 3 
    * The population mean pain relief time is 3 hours.

* **Alternative Hypothesis (H1):** 
    * μ ≠ 3 
    * The population mean pain relief time is not 3 hours.

**2. Given**

* Sample mean (x̄) = 2.5 hours
* Population standard deviation (σ) = 1 hour
* Sample size (n) = 16 patients
* Significance level (α) = 0.05

**3. Calculate the Test Statistic (z-score)**

Since we know the population standard deviation (σ), we can use the z-test:

* z = (x̄ - μ) / (σ / √n)
* z = (2.5 - 3) / (1 / √16) 
* z = -0.5 / 0.25
* z = -2

**4. Determine the Critical Value**

* This is a two-tailed test (since H1 is μ ≠ 3).
* Find the critical z-values for α/2 = 0.05/2 = 0.025 in a standard normal distribution table.
* The critical z-values are approximately ±1.96.

**5. Decision Rule**

* If the calculated z-score falls within the critical region (z < -1.96 or z > 1.96), reject the null hypothesis.
* Otherwise, fail to reject the null hypothesis.

**6. Make a Decision**

* Our calculated z-score (-2) is less than the lower critical value (-1.96).
* Therefore, we **reject the null hypothesis**.

**7. Conclusion**

* At the 0.05 significance level, there is sufficient evidence to conclude that the population mean pain relief time of the new analgesic drug is **different from 3 hours**. 

**In summary:**

The sample data suggests that the average pain relief time for the 16 patients is significantly different from the claimed 3 hours. This could indicate that the drug's effectiveness might vary in the general population. 

**Note:**

* This analysis assumes that the sample is normally distributed or the sample size is large enough for the Central Limit Theorem to apply. 
* Further investigation might be warranted to understand the reasons for this difference.