Question 1171061
Let's conduct the hypothesis test to see if the sample group has a lower performance than the general population.

**1. State the Hypotheses**

* **Null Hypothesis (H₀):** The sample group's mean score is equal to the population mean score.
    * H₀: μ = 85
* **Alternative Hypothesis (H₁):** The sample group's mean score is different from the population mean score.
    * H₁: μ ≠ 85 (two-tailed test)

**2. Determine the Test Statistic**

* We are given the population standard deviation, so we will use a z-test.
* The formula for the z-statistic is:

    z = (x̄ - μ) / (σ / √n)

    Where:
    * x̄ = sample mean
    * μ = population mean
    * σ = population standard deviation
    * n = sample size

**3. Calculate the Test Statistic**

* x̄ = 83.20
* μ = 85
* σ = 8
* n = 50

    z = (83.20 - 85) / (8 / √50)
    z = -1.8 / (8 / 7.071)
    z = -1.8 / 1.131
    z ≈ -1.591

**4. Determine the Critical Value or P-value**

* Significance level (α) = 0.05
* Type of test: Two-tailed

* **Critical Value Approach:**
    * For a two-tailed test with α = 0.05, the critical z-values are ±z<sub>α/2</sub> = ±z<sub>0.025</sub>.
    * Using a standard normal distribution table or a calculator, we find that z<sub>0.025</sub> ≈ 1.96.

* **P-value Approach:**
    * Using a z-table or calculator, we find the p-value associated with z = -1.591 for a two-tailed test.
    * P(Z < -1.591) ≈ 0.0558
    * P(Z > 1.591) ≈ 0.0558
    * p-value = 2 \* 0.0558 ≈ 0.1116

**5. Make a Decision**

* **Critical Value Approach:**
    * The calculated z-statistic is -1.591.
    * The critical z-values are ±1.96.
    * Since |-1.591| < 1.96, we fail to reject the null hypothesis.

* **P-value Approach:**
    * The p-value is approximately 0.1116.
    * The significance level is 0.05.
    * Since the p-value (0.1116) is greater than the significance level (0.05), we fail to reject the null hypothesis.

**6. Conclusion**

* There is not sufficient evidence at the 0.05 significance level to conclude that the sample group has a lower performance than the general population.