One Population 1) In past years the mean IQ of enlisted men in the Army was 103 . A general claims that due to low pay, the average \( 1 Q \) of new enlisted men is lower than in the past. In a random sample of 60 enlistees, the mean \( 1 Q \) was 98 with a standard deviation of 20 . a) At the \( 8 \% \) significance level, what can be concluded about general's claim? The average IQ of hew en listed men is b) What is the p-value of this test?

Let's tackle each part of the problem step-by-step. ### **Part a: Conclusion at the 8% Significance Level** **Given:** - **Population Mean (μ₀):** 103 - **Sample Mean (x̄):** 98 - **Sample Standard Deviation (s):** 20 - **Sample Size (n):** 60 - **Significance Level (α):** 8% (0.08) **Hypotheses:** - **Null Hypothesis (H₀):** μ = 103 - **Alternative Hypothesis (Hₐ):** μ < 103 **Test Statistic Calculation:** We use the z-test since the sample size is large (n > 30). \[ z = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{98 - 103}{20 / \sqrt{60}} = \frac{-5}{2.583} \approx -1.936 \] **Critical Value:** For a left-tailed test at α = 0.08, the critical z-value (z₀.₀₈) is approximately **-1.405**. **Decision:** Since **-1.936 < -1.405**, we reject the null hypothesis. **Conclusion:** There is sufficient evidence at the 8% significance level to support the general's claim that the average IQ of new enlisted men is **lower than 103**. --- ### **Part b: Calculating the p-Value** The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. **Test Statistic:** z ≈ -1.936 **p-Value Calculation:** \[ \text{p-value} = P(Z \leq -1.936) = P(Z \geq 1.936) \approx 0.0264 \text{ or } 2.64\% \] **Interpretation:** A p-value of approximately **2.6%** indicates that there's a 2.6% probability of observing an average IQ of 98 or lower in a sample of 60 enlistees if the true population mean were 103. --- **Final Answers:** **a)** At the 8% significance level, we reject the null hypothesis and conclude that the average IQ of new enlisted men is **lower than 103**. **b)** The p-value of this test is approximately **2.6%**.