SOLUTION: A researcher is interested in the conditions that affect the number of dreams per month that people report in which they are alone. We will assume that based on extensive previous

Question 1185744: A researcher is interested in the conditions that affect the number of dreams per month that people report in which they are alone. We will assume that based on extensive previous research, it is known that in the general population the number of such dreams per month follows a normal curve, with Population M = 5 and Population SD = 4. The researcher wants to test the prediction that the number of such dreams will be greater among people who have recently experienced a traumatic event. Thus, the researcher studies 36 individuals who have recently experienced a traumatic event, having them keep a record of their dreams for a month. Their mean number of alone dreams is 8. SHould you conclude that people who have recently had a traumatic experience have a significantly different number of dreams in which they are alone? (A) Carry out a Z test using the five steps of hypothesis testing (use the .05 significance level) Please include the Population M, Population SD, N, M, one tail or two tail, Population Mm, Population SD2m, Standard Deviation population SDm, Null Hypothesis
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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

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