Question 1200425: A pharmaceutical firm claims that a new analgesic drug relieves mild pain under standard conditions for 3 hours, with a standard deviation 1 hour. Sixteen patients are tested under the same conditions and have an average pain relief time of 2.5 hours. Test the hypothesis that the population mean of this sample is actually 3 hours. Use ( =0.05). Try to follow all steps in hypothesis testing.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! 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.
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