Answer :
Let's go through the steps to estimate the mean, median, and mode for the given data.
### a) Estimate the Mean
1. Identify the class intervals and frequencies:
- Class Intervals: [tex]\[(65, 99.99), (100, 134.99), (135, 169.99), (170, 204.99), (205, 239.99)\][/tex]
- Frequencies: [tex]\[3, 2, 5, 5, 8\][/tex]
2. Calculate the midpoints for each interval:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- For (65, 99.99): [tex]\(\frac{65 + 99.99}{2} = 82.495\)[/tex]
- For (100, 134.99): [tex]\(\frac{100 + 134.99}{2} = 117.495\)[/tex]
- For (135, 169.99): [tex]\(\frac{135 + 169.99}{2} = 152.495\)[/tex]
- For (170, 204.99): [tex]\(\frac{170 + 204.99}{2} = 187.495\)[/tex]
- For (205, 239.99): [tex]\(\frac{205 + 239.99}{2} = 222.495\)[/tex]
3. Calculate the mean using the formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{Midpoint} \times \text{Frequency})}{\sum \text{Frequencies}} \][/tex]
- Sum of (Midpoint × Frequency):
[tex]\[ (82.495 \times 3) + (117.495 \times 2) + (152.495 \times 5) + (187.495 \times 5) + (222.495 \times 8) \][/tex]
- Compute individually:
[tex]\[ 247.485 + 234.99 + 762.475 + 937.475 + 1779.96 = 3962.385 \][/tex]
- Total frequency: [tex]\(3 + 2 + 5 + 5 + 8 = 23\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{3962.385}{23} \approx 172.278 \][/tex]
### b) Estimate the Median
1. Calculate the cumulative frequency:
[tex]\[ \begin{array}{c|c} \text{Class Interval} & \text{Cumulative Frequency} \\ \hline (65, 99.99) & 3 \\ (100, 134.99) & 3 + 2 = 5 \\ (135, 169.99) & 5 + 5 = 10 \\ (170, 204.99) & 10 + 5 = 15 \\ (205, 239.99) & 15 + 8 = 23 \\ \end{array} \][/tex]
2. Locate the median class (where cumulative frequency ≥ total frequency / 2):
- Total frequency [tex]\(\sum \text{Frequencies} = 23\)[/tex]
- Median location [tex]\( = \frac{23}{2} = 11.5\)[/tex] (Look for cumulative frequency ≥ 11.5)
- The median class is (170, 204.99).
3. Calculate the lower boundary, cumulative frequency before the median class, and interval width:
- Lower Boundary (L) = 170
- Cumulative frequency before the median class (F) = 10
- Interval frequency (f) = 5
- Interval width (h) = 204.99 - 170 = 34.99
4. Calculate the median using the formula:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \][/tex]
- Median:
[tex]\[ \text{Median} = 170 + \left( \frac{11.5 - 10}{5} \right) \times 34.99 = 170 + \left(\frac{1.5}{5}\right) \times 34.99 = 170 + 10.497 = 180.497 \][/tex]
### c) Estimate the Mode
1. Locate the modal class (class with the highest frequency):
- The highest frequency is 8. The modal class is (205, 239.99).
2. Calculate the lower boundary (L) and the frequency differences:
- Lower Boundary (L) = 205
- Frequency of modal class ([tex]\(f_m\)[/tex]) = 8
- Frequency of class before modal class ([tex]\(f_1\)[/tex]) = 5
- Frequency of class after modal class ([tex]\(f_2\)[/tex]) = 0
- Interval width (h) = 239.99 - 205 = 34.99
3. Calculate the mode using the formula:
[tex]\[ \text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times h \][/tex]
- Mode:
[tex]\[ \text{Mode} = 205 + \left( \frac{8 - 5}{(8 - 5) + (8 - 0)} \right) \times 34.99 = 205 + \left( \frac{3}{3 + 8} \right) \times 34.99 = 205 + \left( \frac{3}{11} \right) \times 34.99 = 205 + 9.543 = 214.543 \][/tex]
### Summary
- Mean: [tex]\(172.278\)[/tex]
- Median: [tex]\(180.497\)[/tex]
- Mode: [tex]\(214.543\)[/tex]
### a) Estimate the Mean
1. Identify the class intervals and frequencies:
- Class Intervals: [tex]\[(65, 99.99), (100, 134.99), (135, 169.99), (170, 204.99), (205, 239.99)\][/tex]
- Frequencies: [tex]\[3, 2, 5, 5, 8\][/tex]
2. Calculate the midpoints for each interval:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- For (65, 99.99): [tex]\(\frac{65 + 99.99}{2} = 82.495\)[/tex]
- For (100, 134.99): [tex]\(\frac{100 + 134.99}{2} = 117.495\)[/tex]
- For (135, 169.99): [tex]\(\frac{135 + 169.99}{2} = 152.495\)[/tex]
- For (170, 204.99): [tex]\(\frac{170 + 204.99}{2} = 187.495\)[/tex]
- For (205, 239.99): [tex]\(\frac{205 + 239.99}{2} = 222.495\)[/tex]
3. Calculate the mean using the formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{Midpoint} \times \text{Frequency})}{\sum \text{Frequencies}} \][/tex]
- Sum of (Midpoint × Frequency):
[tex]\[ (82.495 \times 3) + (117.495 \times 2) + (152.495 \times 5) + (187.495 \times 5) + (222.495 \times 8) \][/tex]
- Compute individually:
[tex]\[ 247.485 + 234.99 + 762.475 + 937.475 + 1779.96 = 3962.385 \][/tex]
- Total frequency: [tex]\(3 + 2 + 5 + 5 + 8 = 23\)[/tex]
- Mean:
[tex]\[ \text{Mean} = \frac{3962.385}{23} \approx 172.278 \][/tex]
### b) Estimate the Median
1. Calculate the cumulative frequency:
[tex]\[ \begin{array}{c|c} \text{Class Interval} & \text{Cumulative Frequency} \\ \hline (65, 99.99) & 3 \\ (100, 134.99) & 3 + 2 = 5 \\ (135, 169.99) & 5 + 5 = 10 \\ (170, 204.99) & 10 + 5 = 15 \\ (205, 239.99) & 15 + 8 = 23 \\ \end{array} \][/tex]
2. Locate the median class (where cumulative frequency ≥ total frequency / 2):
- Total frequency [tex]\(\sum \text{Frequencies} = 23\)[/tex]
- Median location [tex]\( = \frac{23}{2} = 11.5\)[/tex] (Look for cumulative frequency ≥ 11.5)
- The median class is (170, 204.99).
3. Calculate the lower boundary, cumulative frequency before the median class, and interval width:
- Lower Boundary (L) = 170
- Cumulative frequency before the median class (F) = 10
- Interval frequency (f) = 5
- Interval width (h) = 204.99 - 170 = 34.99
4. Calculate the median using the formula:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \][/tex]
- Median:
[tex]\[ \text{Median} = 170 + \left( \frac{11.5 - 10}{5} \right) \times 34.99 = 170 + \left(\frac{1.5}{5}\right) \times 34.99 = 170 + 10.497 = 180.497 \][/tex]
### c) Estimate the Mode
1. Locate the modal class (class with the highest frequency):
- The highest frequency is 8. The modal class is (205, 239.99).
2. Calculate the lower boundary (L) and the frequency differences:
- Lower Boundary (L) = 205
- Frequency of modal class ([tex]\(f_m\)[/tex]) = 8
- Frequency of class before modal class ([tex]\(f_1\)[/tex]) = 5
- Frequency of class after modal class ([tex]\(f_2\)[/tex]) = 0
- Interval width (h) = 239.99 - 205 = 34.99
3. Calculate the mode using the formula:
[tex]\[ \text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times h \][/tex]
- Mode:
[tex]\[ \text{Mode} = 205 + \left( \frac{8 - 5}{(8 - 5) + (8 - 0)} \right) \times 34.99 = 205 + \left( \frac{3}{3 + 8} \right) \times 34.99 = 205 + \left( \frac{3}{11} \right) \times 34.99 = 205 + 9.543 = 214.543 \][/tex]
### Summary
- Mean: [tex]\(172.278\)[/tex]
- Median: [tex]\(180.497\)[/tex]
- Mode: [tex]\(214.543\)[/tex]
