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Let's break down this question.

**Part a)**:
We are given two students:
- Anna scored 84 on both exams.
- Megan scored 74 on the first exam and 91 on the second exam.

The overall scores for all students in these exams are:
- First exam: mean = 81, standard deviation = 7.
- Second exam: mean = 71, standard deviation = 14.

To qualify for language honors, the requirement is a **mean score of at least 85** across all language courses.

Now, let's calculate the **mean scores** for both Anna and Megan:
- **Anna**: Her scores are 84 and 84. Her average = (84 + 84) / 2 = **84**.
- **Megan**: Her scores are 74 and 91. Her average = (74 + 91) / 2 = **82.5**.

Since the required average is **85**, neither Anna (average = 84) nor Megan (average = 82.5) qualifies for honors. 

Thus, the answer for part (a) is **C: Neither Anna nor Megan qualify**.

---

**Part b)**:
To determine which student performed better overall, we can consider the **z-scores** for each student's performance relative to the class performance. 

The z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(X\) is the student's score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

For **Anna**:
- For the first exam: \(Z = \frac{84 - 81}{7} = \frac{3}{7} \approx 0.43\).
- For the second exam: \(Z = \frac{84 - 71}{14} = \frac{13}{14} \approx 0.93\).

For **Megan**:
- For the first exam: \(Z = \frac{74 - 81}{7} = \frac{-7}{7} = -1\).
- For the second exam: \(Z = \frac{91 - 71}{14} = \frac{20}{14} \approx 1.43\).

Now, let's average the z-scores:
- Anna: \(Z_{avg} = \frac{0.43 + 0.93}{2} = 0.68\).
- Megan: \(Z_{avg} = \frac{-1 + 1.43}{2} = 0.215\).

Since **Anna's z-score is higher**, **Anna performed better overall**.

Thus, the answer for part (b) is **Anna performed better overall**.

Would you like any further details or explanations?

---

Here are 5 related questions to explore:

1. How do z-scores help compare performances across different distributions?
2. What are the steps to calculate the standard deviation from a set of scores?
3. If Megan had scored 88 on her second exam, would she qualify for honors?
4. How can you use mean and standard deviation to assess student performance?
5. What is the significance of z-scores in statistics?

**Tip**: Always compare using z-scores when the scales of measurement or the variability in data differ across categories.

Two language majors, Anna and Megan, took exams in two languages. Anna scored 84 on both exams. Megan scored 74 on the first exam and 91 on the second exam. Overall, student scores on the first exam had a mean of 81 and a standard deviation of 7, and the second exam scores had a mean of 71 and a standard deviation of 14. a) To qualify for language honors, a major must maintain at least an 85 average across all language courses taken. So far, which of Anna and Megan qualify? b) Which student's overall performance was better?

Analyze Anna and Megan's exam scores to determine if they qualify for language honors and compare their overall performance using mean and z-scores. This problem explores key statistical concepts such as mean, standard deviation, and z-scores, suitable for college-level statistics.

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