Question 1185356
Here's how to conduct a hypothesis test to determine if there's a significant improvement in patient satisfaction:

**1. State the Hypotheses:**

*   **Null Hypothesis (H₀):** Patient satisfaction has not increased (or has decreased). The mean satisfaction score in 2003 is less than or equal to the mean satisfaction score in 2002. μ ≤ 84.5
*   **Alternative Hypothesis (H₁):** Patient satisfaction has increased. The mean satisfaction score in 2003 is greater than the mean satisfaction score in 2002. μ > 84.5 (This is a one-tailed test).

**2. Determine the Significance Level:**

α = 0.01

**3. Choose the appropriate test statistic:**

Since the sample size is large (n = 125 > 30) and the population standard deviation is unknown, we can use a one-sample z-test.  While a t-test would also be appropriate, with such a large sample, the results will be very similar.

**4. Calculate the test statistic:**

The z-statistic is calculated as:

z = (sample mean - population mean) / (sample standard deviation / √sample size)
z = (89.2 - 84.5) / (17.4 / √125)
z = 4.7 / (17.4 / 11.18)
z = 4.7 / 1.556
z ≈ 3.02

**5. Determine the critical value (or p-value):**

*   **Using a z-table:** For a one-tailed test with α = 0.01, the critical z-value is approximately 2.33.
*   **Using a calculator or statistical software:** A calculator or statistical software can provide a more precise p-value.

**6. Calculate the p-value:**

Using statistical software or a z-table, with z ≈ 3.02, the p-value is very small, much less than 0.01 (approximately 0.0013).

**7. Make a decision:**

*   **Using the critical value:** Our calculated z-statistic (3.02) is greater than the critical z-value (2.33). Therefore, we reject the null hypothesis.
*   **Using the p-value:** The p-value (≈ 0.0013) is less than the significance level (0.01). Therefore, we reject the null hypothesis.

**8. Conclusion:**

There is very strong evidence at the α = 0.01 level of significance to conclude that the quality-improvement initiatives have increased patient satisfaction.