Based on the bar chart showing the $95\%$ confidence intervals for the mean test scores of four different classes, which of the following is an accurate conclusion?

• A. A wider confidence interval always indicates a higher mean test score.
• B. If the confidence intervals for Class C and Class D overlap, it means their mean test scores are exactly equal.
• C. If the confidence intervals for Class A and Class B do not overlap, we can conclude that there is a statistically significant difference between their mean test scores at the $5\%$ significance level.
• D. If the confidence interval for a class includes $0$, it means the mean test score for that class is $0$.

Answer: (C)

Verified step by step guidance

1. Understand what a 95% confidence interval represents: it is a range of values within which we are 95% confident that the true population mean lies.
2. Recognize that the width of a confidence interval depends on the variability of the data and the sample size, not directly on the mean value itself. Therefore, a wider confidence interval does not necessarily indicate a higher mean test score.
3. Know that overlapping confidence intervals between two groups (e.g., Class C and Class D) do not imply that their means are exactly equal; rather, it suggests that there is not enough evidence to say the means are different at the 5% significance level.
4. Understand that if the 95% confidence intervals for two classes (e.g., Class A and Class B) do not overlap, this is strong evidence that their mean test scores are statistically significantly different at the 5% significance level.
5. Remember that a confidence interval including zero means that zero is a plausible value for the population mean, but it does not mean the mean test score is exactly zero.