1. **Stating the problem:**
We have a histogram and partial frequency table for heights of Year 11 students. We need to:
a) Complete the frequency table using the histogram.
b) Complete the histogram using the frequency table.
2. **Understanding frequency density and frequency:**
Frequency density = \frac{Frequency}{Class width}
Frequency = Frequency density \times Class width
3. **Given data:**
- Heights intervals and some frequencies:
- 140 ≤ h < 150: Frequency = 15
- 150 ≤ h < 160: Frequency = ?
- 160 ≤ h < 165: Frequency = 20
- 165 ≤ h < 170: Frequency = ?
- 170 ≤ h < 180: Frequency = ?
- 180 ≤ h < 190: Frequency = ?
- 190 ≤ h < 210: Frequency = 12
- Histogram bar heights (frequency densities) approximately:
- 140 ≤ h < 150: 3
- 150 ≤ h < 160: 7.5
- 165 ≤ h < 170: 8
- 170 ≤ h < 180: 4.5
- 190 ≤ h < 210: 1.2
4. **Calculate class widths:**
- 140 ≤ h < 150: width = 10
- 150 ≤ h < 160: width = 10
- 160 ≤ h < 165: width = 5
- 165 ≤ h < 170: width = 5
- 170 ≤ h < 180: width = 10
- 180 ≤ h < 190: width = 10
- 190 ≤ h < 210: width = 20
5. **Calculate missing frequencies using frequency density and class width:**
- For 140 ≤ h < 150:
Frequency density = 3
Frequency = 3 × 10 = 30 (But given frequency is 15, so frequency density must be recalculated)
- Recalculate frequency density for 140 ≤ h < 150:
Frequency density = \frac{15}{10} = 1.5
- For 150 ≤ h < 160:
Frequency density = 7.5
Frequency = 7.5 × 10 = 75
- For 160 ≤ h < 165:
Frequency = 20 (given)
Frequency density = \frac{20}{5} = 4
- For 165 ≤ h < 170:
Frequency density = 8
Frequency = 8 × 5 = 40
- For 170 ≤ h < 180:
Frequency density = 4.5
Frequency = 4.5 × 10 = 45
- For 180 ≤ h < 190:
Frequency density is missing, frequency is missing
Use frequency density from histogram (not given), but we can find frequency density using frequency if known.
- For 190 ≤ h < 210:
Frequency density = 1.2
Frequency = 1.2 × 20 = 24 (But given frequency is 12, so frequency density must be recalculated)
- Recalculate frequency density for 190 ≤ h < 210:
Frequency density = \frac{12}{20} = 0.6
6. **Summary of frequency densities and frequencies:**
| Interval | Width | Frequency Density | Frequency |
|----------------|-------|-------------------|-----------|
| 140 ≤ h < 150 | 10 | 1.5 | 15 |
| 150 ≤ h < 160 | 10 | 7.5 | 75 |
| 160 ≤ h < 165 | 5 | 4 | 20 |
| 165 ≤ h < 170 | 5 | 8 | 40 |
| 170 ≤ h < 180 | 10 | 4.5 | 45 |
| 180 ≤ h < 190 | 10 | ? | ? |
| 190 ≤ h < 210 | 20 | 0.6 | 12 |
7. **Find missing frequency density and frequency for 180 ≤ h < 190:**
From the histogram, the bar for 180 ≤ h < 190 is missing, but since frequency is missing, we can estimate frequency density by interpolation or assume it is similar to nearby bars.
Assuming frequency density for 180 ≤ h < 190 is approximately 3 (estimated from histogram pattern), then:
Frequency = 3 × 10 = 30
8. **Final completed frequency table:**
| Height (h cm) | Frequency |
|----------------|-----------|
| 140 ≤ h < 150 | 15 |
| 150 ≤ h < 160 | 75 |
| 160 ≤ h < 165 | 20 |
| 165 ≤ h < 170 | 40 |
| 170 ≤ h < 180 | 45 |
| 180 ≤ h < 190 | 30 |
| 190 ≤ h < 210 | 12 |
9. **To complete the histogram:**
Use frequency densities calculated as frequency ÷ class width for each interval.
10. **Answer:**
- Completed frequency table as above.
- Completed histogram bars with frequency densities:
- 140 ≤ h < 150: 1.5
- 150 ≤ h < 160: 7.5
- 160 ≤ h < 165: 4
- 165 ≤ h < 170: 8
- 170 ≤ h < 180: 4.5
- 180 ≤ h < 190: 3 (estimated)
- 190 ≤ h < 210: 0.6
This completes the problem.
Histogram Frequency 127Bc2
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