Question 1170393
Here's how to approach this hypothesis test:

**1. Define the Hypotheses:**

* **Null Hypothesis (H0):** The actual prevalence of GAD among children and adolescents is 3.9% (p = 0.039).
* **Alternative Hypothesis (H1):** The actual prevalence of GAD among children and adolescents is higher than 3.9% (p > 0.039).

**2. Set the Significance Level:**

* α = 0.05 (5% significance level)

**3. Calculate the Sample Proportion:**

* Sample size (n) = 98
* Number of children with GAD = 98 * 0.061 = 5.978. Since you cant have a fraction of a child, we will round to 6.
* Sample proportion (p̂) = 6 / 98 ≈ 0.0612

**4. Perform the Hypothesis Test (One-Proportion Z-Test):**

* We'll use a one-proportion z-test because we're dealing with proportions.
* The test statistic (z) is calculated as:

    $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

    Where:
    * p̂ = sample proportion (0.0612)
    * p = population proportion (0.039)
    * n = sample size (98)

* Now we plug in the values:

    $z = \frac{0.0612 - 0.039}{\sqrt{\frac{0.039(1-0.039)}{98}}}$

    $z = \frac{0.0222}{\sqrt{\frac{0.039(0.961)}{98}}}$

    $z = \frac{0.0222}{\sqrt{\frac{0.03748}{98}}}$

    $z = \frac{0.0222}{\sqrt{0.0003824}}$

    $z = \frac{0.0222}{0.01955}$

    $z ≈ 1.135$

**5. Determine the Critical Value or P-value:**

* Since this is a right-tailed test (H1: p > 0.039), we need to find the critical z-value for α = 0.05.
* Using a standard normal distribution table or calculator, the critical z-value is approximately 1.645.
* Alternativly we can calculate the p value.
* Using a z value of 1.135, and a right tailed test, the p value is .128.

**6. Make a Decision:**

* **Critical Value Method:**
    * Our calculated z-value (1.135) is less than the critical z-value (1.645).
    * Therefore, we fail to reject the null hypothesis.

* **P-value method:**
    * The p value of .128 is larger than the significance level of .05.
    * Therefore, we fail to reject the null hypothesis.

**7. Draw a Conclusion:**

* At the 5% significance level, there is not enough evidence to support the researcher's claim that the actual prevalence of GAD among children and adolescents is higher than 3.9%.
* The researcher should conclude that the difference between the sample proportion and the previously reported proportion is not statistically significant.

**In summary:** The researcher does not have sufficient evidence to reject the previously reported prevalence.