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 IQ of new enlisted men is lower than in the past. In a random sample of 60 enlistees, the mean IQ was 98 with a standard deviation of 20 . a) At the \( 8 \% \) significance level, what can be concluded about general's claim? b) What is the p-value of this test?
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To determine whether the general's claim that the average IQ of new enlisted men is lower than in the past is statistically significant, we will conduct a one-sample t-test. a) Null Hypothesis (\(H_0\)): The mean IQ of new enlisted men is 103. Alternative Hypothesis (\(H_a\)): The mean IQ of new enlisted men is less than 103. Using the formula for the t statistic: \[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{98 - 103}{20 / \sqrt{60}} \approx \frac{-5}{2.58} \approx -1.94 \] Degrees of freedom (\(df\)) = \(n-1 = 60-1 = 59\). We compare the t value to critical t values for a one-tailed test at \( \alpha = 0.08\), which is approximately -1.41 (using a t-table). Since -1.94 < -1.41, we reject the null hypothesis. This suggests that there is statistically significant evidence at the 8% level to support the general's claim that the average IQ of new enlisted men is lower than in the past. b) To find the p-value, we can look at the t-distribution with 59 degrees of freedom. Using a t-table or calculator: for \( t = -1.94 \), the p-value is approximately 0.03. Since 0.03 < 0.08, it supports our conclusion to reject the null hypothesis, reinforcing the general's claim.
