A school administrator wants to examine whether students' academic performance differs based on the type of instructional method used in their classes. A random sample of $18$ students is selected and divided evenly among the three teaching methods. After a semester, all students take the same standardized final exam. An ANOVA test is performed and results in a P-value of $1.403\bullet10^{-7}$. Interpret these results.

• A. The P-value is very high, so there is insufficient evidence to suggest that the type of instructional method has an effect on academic performance.
• B. At least one method has a different average test score. Reject the null hypothesis.
• C. There is no difference in academic performance, but further testing is required to make a definitive conclusion.
• D.  The P-value is close to 1, suggesting that the type of instructional method has no impact on students' academic performance.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. The school administrator wants to determine if the type of instructional method (Traditional, Flipped, Online) affects students' academic performance. An ANOVA test is used to compare the means of test scores across the three groups.
2. Step 2: Recall the null and alternative hypotheses for an ANOVA test. The null hypothesis (H₀) states that all group means are equal, i.e., the type of instructional method has no effect on academic performance. The alternative hypothesis (H₁) states that at least one group mean is different.
3. Step 3: Examine the P-value provided in the problem. The P-value is 1.403 × 10⁻⁷, which is extremely small. In hypothesis testing, if the P-value is less than the significance level (commonly α = 0.05), we reject the null hypothesis.
4. Step 4: Interpret the results. Since the P-value is much smaller than 0.05, we reject the null hypothesis. This indicates that there is sufficient evidence to conclude that at least one instructional method has a different average test score compared to the others.
5. Step 5: Relate the conclusion to the data. The test scores provided in the table show noticeable differences in averages between the methods (e.g., Flipped scores are generally higher than Traditional and Online). This supports the statistical conclusion that the type of instructional method impacts academic performance.