SOLUTION: 4. A random sample of 200 school managers were administered a developed Leadership Skills Test. The sample mean and the standard deviation were 78 and 4.2 respectively. In the stan

Question 1186603: 4. A random sample of 200 school managers were administered a developed Leadership Skills Test. The sample mean and the standard deviation were 78 and 4.2 respectively. In the standardization of the test, the mean was 73 and the standard deviation was 8. Test for significant difference using alpha = 0.05 utilizing the p-value method.
answer the problem using the steps in hypothesis testing.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to perform a hypothesis test for this situation:
**1. State the Hypotheses:**
* **Null Hypothesis (H₀):** There is no significant difference between the sample mean and the population mean. μ = 73
* **Alternative Hypothesis (H₁ or Ha):** There is a significant difference between the sample mean and the population mean. μ ≠ 73 (This is a two-tailed test.)
**2. Determine the Level of Significance (alpha):**
* α = 0.05 (This is given in the problem.)
**3. Calculate the Test Statistic:**
* Since the sample size is large (n = 200), we can use a z-test.
* **Sample Mean (x̄):** x̄ = 78
* **Population Mean (μ):** μ = 73
* **Population Standard Deviation (σ):** σ = 8
* **Standard Error (SE):** SE = σ / sqrt(n) = 8 / sqrt(200) ≈ 0.566
* **Z-score:** z = (x̄ - μ) / SE = (78 - 73) / 0.566 ≈ 8.83
**4. Determine the P-value:**
* Since this is a two-tailed test, we need to find the probability of getting a z-score as extreme as 8.83 in *either* direction.
* Given a z-score of 8.83, the p-value will be extremely small, essentially close to 0. Most statistical tables won't even list such a high z-score.
**5. Make a Decision:**
* **Compare the p-value to alpha:** The p-value (≈ 0) is *much less* than the level of significance (0.05).
* **Decision:** Because the p-value is less than alpha, we reject the null hypothesis.
**6. Conclusion:**
There is very strong evidence at the 0.05 level of significance to conclude that there is a significant difference between the mean score of the school managers and the population mean. The sample data suggests that the school managers have a significantly higher mean score on the Leadership Skills Test than the population used for standardization.

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