Question 1198525: The administrator of a local hospital has told the governing board that 30% of its
emergency room patients are not really in need of emergency treatment. In checking
a random sample of 400 emergency room patients, a board member finds that 35%
of those treated were not true emergency cases. Using an appropriate hypothesis test
and the 0.05 level, evaluate the administrator’s statement.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Set up Hypotheses**
* **Null Hypothesis (H₀):** p = 0.30 (The true proportion of patients not needing emergency treatment is 30%)
* **Alternative Hypothesis (H₁):** p ≠ 0.30 (The true proportion of patients not needing emergency treatment is different from 30%)
**2. Calculate Sample Proportion**
* Sample Proportion (p̂): 35% = 0.35
**3. Calculate Test Statistic (z-score)**
* z = (p̂ - p₀) / √[p₀ * (1 - p₀) / n]
* where:
* p̂ = sample proportion (0.35)
* p₀ = hypothesized population proportion (0.30)
* n = sample size (400)
* z = (0.35 - 0.30) / √[0.30 * (1 - 0.30) / 400]
* z = 0.05 / √[0.30 * 0.70 / 400]
* z = 0.05 / √[0.21 / 400]
* z = 0.05 / 0.0229
* z ≈ 2.18
**4. Determine Critical Values**
* Since this is a two-tailed test (H₁: p ≠ 0.30) at a 0.05 significance level, we need to find the critical values that divide the distribution into two tails with 0.025 area in each.
* Using a standard normal distribution table, the critical values are approximately ±1.96.
**5. Make a Decision**
* **Compare the test statistic to the critical values:**
* Our calculated z-score (2.18) is greater than the critical value (1.96).
* **Decision:** Since the test statistic falls in the rejection region, we **reject the null hypothesis**.
**6. Conclusion**
* There is sufficient evidence at the 0.05 significance level to reject the administrator's claim that 30% of emergency room patients are not true emergencies. The sample data suggests that the actual proportion may be higher.
**In Summary:**
The board member's findings suggest that a higher proportion of patients may not be true emergencies than the administrator's claim. This information can be valuable for hospital resource allocation and improving emergency room efficiency.
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