Question 1185744
Here's the Z-test, following the five steps of hypothesis testing:

**(A) Z-test for Alone Dreams After Trauma**

**1. State the Hypotheses:**

*   **Null Hypothesis (H0):** There is no difference in the number of alone dreams between people who have experienced trauma and the general population.  μ ≤ 5
*   **Alternative Hypothesis (H1):** People who have experienced trauma have *more* alone dreams. μ > 5 (This is a one-tailed, right-tailed test).

**2. Set the Criteria for a Decision:**

*   **Significance Level (alpha):** α = 0.05
*   **Critical Value:** Since it's a one-tailed test, we look up the z-score corresponding to 0.05 in the right tail of the standard normal distribution. This critical value is approximately 1.645.
*   **Decision Rule:** Reject H0 if the calculated z-score is greater than 1.645.

**3. Compute the Test Statistic:**

```
z = (M - μ) / σM
```

Where:

*   M = Sample mean = 8
*   μ = Population mean = 5
*   σM = Standard error of the mean = σ / √N
*   σ = Population standard deviation = 4
*   N = Sample size = 36

First, calculate the standard error of the mean (σM):

```
σM = 4 / √36 = 4 / 6 = 0.67 (approximately)
```

Now, calculate the z-score:

```
z = (8 - 5) / 0.67
z ≈ 4.48
```

**4. Make a Decision:**

*   **Comparison:** The calculated z-score (4.48) is *much greater* than the critical value (1.645).
*   **Decision:** We *reject* the null hypothesis.

**5. State the Conclusion:**

There is sufficient evidence at the 0.05 significance level to conclude that people who have recently experienced a traumatic event have a significantly greater number of alone dreams per month than the general population.

**Summary of Values:**

*   Population M (μ): 5
*   Population SD (σ): 4
*   N: 36
*   M (Sample Mean): 8
*   One-tailed or Two-tailed: One-tailed (right-tailed)
*   Population Mean of the Sampling Distribution (μM): 5 (same as population mean)
*   Population Variance of the Sampling Distribution (σ²M): σ²/N = 4²/36 = 16/36 = 4/9 ≈ 0.44
*   Standard Deviation of the Sampling Distribution (σM): σ/√N = 4/√36 = 4/6 = 2/3 ≈ 0.67
*   Null Hypothesis: μ ≤ 5