Answer :
Let's solve each part of the question step by step.
### (a) Draw a Histogram
To draw a histogram, we'll represent the given data by plotting the mass intervals (classes) on the x-axis and the frequency of each interval on the y-axis.
The data is as follows:
- Mass of grade 6 boys: [36-40], [41-45], [46-50], [51-55], [56-60], [61-65-70]
- Frequency: [3, 8, 12, 20, 10, 6]
Here is a step-by-step guide to create the histogram:
1. X-Axis (Mass Intervals):
- Plot the intervals (36-40, 41-45, 46-50, 51-55, 56-60, 61-65-70).
2. Y-Axis (Frequency):
- Plot the corresponding frequencies for each interval (3, 8, 12, 20, 10, 6).
3. Bars:
- For each mass interval on the x-axis, draw a bar that extends to the corresponding frequency on the y-axis.
### (b) Draw a Cumulative Frequency Curve
To draw a cumulative frequency curve, we'll calculate the cumulative frequency and then plot it against the midpoints of each class interval.
#### Midpoints of the Mass Intervals
The midpoint for each interval is calculated as follows:
- For 36-40: [tex]\( \frac{36 + 40}{2} = 38 \)[/tex]
- For 41-45: [tex]\( \frac{41 + 45}{2} = 43 \)[/tex]
- For 46-50: [tex]\( \frac{46 + 50}{2} = 48 \)[/tex]
- For 51-55: [tex]\( \frac{51 + 55}{2} = 53 \)[/tex]
- For 56-60: [tex]\( \frac{56 + 60}{2} = 58 \)[/tex]
- For 61-65-70: Using 63 (averging first 3 point) or alternative is using the midpoint of 61-70 range \[ \text{ both have needs clarify for academic view but lets start with} [tex]\( \frac{61 + 65}{2} = 63 \)[/tex]
#### Cumulative Frequencies
Cumulative frequency at each interval is the sum of all previous frequencies up to that interval:
- Cumulative frequency for 36-40: 3
- Cumulative frequency for 41-45: [tex]\( 3 + 8 = 11 \)[/tex]
- Cumulative frequency for 46-50: [tex]\( 11 + 12 = 23 \)[/tex]
- Cumulative frequency for 51-55: [tex]\( 23 + 20 = 43 \)[/tex]
- Cumulative frequency for 56-60: [tex]\( 43 + 10 = 53 \)[/tex]
- Cumulative frequency for 61-65-70: [tex]\( 53 + 6 = 59 \)[/tex]
### (c) Use the Cumulative Frequency Curve to Estimate:
To find the median, 25th percentile, 75th percentile, and semi-interquartile range, follow these steps:
1. Median (50th percentile):
- Total number of students: 59
- Median position: [tex]\( \frac{59 + 1}{2} = 30 \)[/tex]
- Identify the interval containing the 30th student: it's in the 51-55 interval.
- Exact calculation: Using interpolation within the interval.
\[ \text {Approx median} =52 or using by closer calculation\approx 52.5
2. 25th percentile (First Quartile):
- 25th percentile position: [tex]\( 0.25 \times 59 = 14.75 \)[/tex]
- Identify the interval: it's in the 46-50 interval.
- Using interpolation within the interval:[as middle/closer 24]
- The approx value at 14.75=47.5
\approx 46.5
3. 75th percentile (Third Quartile):
- 75th percentile position: [tex]\( 0.75 \times 59 = 44.25\ or 42\approx0.75 = 44.25 - Identify the interval: it's in the 56-60 interval. - Using interpolation within the interval:[as middle/closer 25] -elta value to measure shift within approximation - Approx value at 70 approx dust =67.5 approx 4. Semi-Interquartile Range: - Semi-interquartile range: \( \frac{75th\ percentile\ -\ 25th percentile}{2} \)[/tex]
- Semi-interquartile range: \( \frac{67.5-47.5}{2}\approx 10 [endale otherwise centers \approx15]
Thus, we found midpoints to ensure better fit in class interval. Therefore step wise measure also visually histogram and cumulative freq enforces estimate:
Steps through data clearly, reliably and visually noting quartiles boundaries and semi-interquartile.
### (a) Draw a Histogram
To draw a histogram, we'll represent the given data by plotting the mass intervals (classes) on the x-axis and the frequency of each interval on the y-axis.
The data is as follows:
- Mass of grade 6 boys: [36-40], [41-45], [46-50], [51-55], [56-60], [61-65-70]
- Frequency: [3, 8, 12, 20, 10, 6]
Here is a step-by-step guide to create the histogram:
1. X-Axis (Mass Intervals):
- Plot the intervals (36-40, 41-45, 46-50, 51-55, 56-60, 61-65-70).
2. Y-Axis (Frequency):
- Plot the corresponding frequencies for each interval (3, 8, 12, 20, 10, 6).
3. Bars:
- For each mass interval on the x-axis, draw a bar that extends to the corresponding frequency on the y-axis.
### (b) Draw a Cumulative Frequency Curve
To draw a cumulative frequency curve, we'll calculate the cumulative frequency and then plot it against the midpoints of each class interval.
#### Midpoints of the Mass Intervals
The midpoint for each interval is calculated as follows:
- For 36-40: [tex]\( \frac{36 + 40}{2} = 38 \)[/tex]
- For 41-45: [tex]\( \frac{41 + 45}{2} = 43 \)[/tex]
- For 46-50: [tex]\( \frac{46 + 50}{2} = 48 \)[/tex]
- For 51-55: [tex]\( \frac{51 + 55}{2} = 53 \)[/tex]
- For 56-60: [tex]\( \frac{56 + 60}{2} = 58 \)[/tex]
- For 61-65-70: Using 63 (averging first 3 point) or alternative is using the midpoint of 61-70 range \[ \text{ both have needs clarify for academic view but lets start with} [tex]\( \frac{61 + 65}{2} = 63 \)[/tex]
#### Cumulative Frequencies
Cumulative frequency at each interval is the sum of all previous frequencies up to that interval:
- Cumulative frequency for 36-40: 3
- Cumulative frequency for 41-45: [tex]\( 3 + 8 = 11 \)[/tex]
- Cumulative frequency for 46-50: [tex]\( 11 + 12 = 23 \)[/tex]
- Cumulative frequency for 51-55: [tex]\( 23 + 20 = 43 \)[/tex]
- Cumulative frequency for 56-60: [tex]\( 43 + 10 = 53 \)[/tex]
- Cumulative frequency for 61-65-70: [tex]\( 53 + 6 = 59 \)[/tex]
### (c) Use the Cumulative Frequency Curve to Estimate:
To find the median, 25th percentile, 75th percentile, and semi-interquartile range, follow these steps:
1. Median (50th percentile):
- Total number of students: 59
- Median position: [tex]\( \frac{59 + 1}{2} = 30 \)[/tex]
- Identify the interval containing the 30th student: it's in the 51-55 interval.
- Exact calculation: Using interpolation within the interval.
\[ \text {Approx median} =52 or using by closer calculation\approx 52.5
2. 25th percentile (First Quartile):
- 25th percentile position: [tex]\( 0.25 \times 59 = 14.75 \)[/tex]
- Identify the interval: it's in the 46-50 interval.
- Using interpolation within the interval:[as middle/closer 24]
- The approx value at 14.75=47.5
\approx 46.5
3. 75th percentile (Third Quartile):
- 75th percentile position: [tex]\( 0.75 \times 59 = 44.25\ or 42\approx0.75 = 44.25 - Identify the interval: it's in the 56-60 interval. - Using interpolation within the interval:[as middle/closer 25] -elta value to measure shift within approximation - Approx value at 70 approx dust =67.5 approx 4. Semi-Interquartile Range: - Semi-interquartile range: \( \frac{75th\ percentile\ -\ 25th percentile}{2} \)[/tex]
- Semi-interquartile range: \( \frac{67.5-47.5}{2}\approx 10 [endale otherwise centers \approx15]
Thus, we found midpoints to ensure better fit in class interval. Therefore step wise measure also visually histogram and cumulative freq enforces estimate:
Steps through data clearly, reliably and visually noting quartiles boundaries and semi-interquartile.
