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The image you've uploaded presents a question about z-scores and how they provide information that non-standardized heights do not. To answer this, let's break down the problem and the options:

### Provided Information:
- **Heights of women (ages 20-29)**: Mean = 64.1 inches, Standard deviation = 3.7 inches.
- **Heights of men (ages 20-29)**: Mean = 69.4 inches, Standard deviation = 3.1 inches.

### Z-Score Explanation:
A **z-score** tells you how many standard deviations a value is from the mean. Z-scores allow for the comparison of two data points from different distributions, in this case, the heights of men and women. 

### Now, analyzing the options:
1. **Option 1**: "The z-scores show us that the woman is slightly taller than average, while the man is much shorter than average."
   - This seems inaccurate based on typical distributions for heights, especially because we have not yet calculated the z-scores.

2. **Option 2**: "The z-scores show that the woman is slightly different from the average, while the man is very different from the average."
   - This may be plausible if z-scores indicate that one individual is closer to their gender mean height than the other. However, "slightly" and "very different" would need calculation of the z-scores to confirm.

3. **Option 3**: "The z-scores do not give us any additional information since we already know that the man and woman have equal heights."
   - Incorrect, since men and women do not have equal heights, and z-scores always provide a deeper understanding of how far a value deviates from the mean.

4. **Option 4**: "The z-scores show us that the woman is slightly shorter than average, while the man is much taller than average."
   - This option seems more reasonable because it aligns with typical distributions (e.g., most women are shorter than men, and men's average height is higher).

### Conclusion:
The correct answer is **Option 4**, as z-scores will likely show that the woman is slightly shorter than average for women, while the man is much taller than average for men.

Do you need help calculating the z-scores, or do you have any other questions about z-scores?

### Follow-up questions:
1. How do you calculate a z-score for a given height?
2. Can you compare z-scores of two different distributions (men and women) directly?
3. What does it mean if someone has a negative z-score?
4. What is the significance of a z-score of 0?
5. How does the standard deviation affect the z-score?

**Tip:** Z-scores are useful when comparing different groups because they account for differences in both means and variability (standard deviations).

What information do the z-scores give that the original non-standardized heights do not?

Explore how z-scores provide additional insights when comparing the heights of men and women, showing deviations from the mean using standard deviations. This statistical problem is perfect for students in Grades 9-12.

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