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To calculate an estimate for the mean height of the students, we'll follow a detailed, step-by-step approach:
### Step-by-Step Solution
1. Identify the Height Intervals and the Frequencies:
The given height intervals and their corresponding frequencies are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Height range (cm)} & \text{Frequency} \\ \hline 120 < h \leqslant 124 & 7 \\ \hline 124 < h \leqslant 128 & 8 \\ \hline 128 < h \leqslant 132 & 13 \\ \hline 132 < h \leqslant 136 & 9 \\ \hline 136 < h \leqslant 140 & 3 \\ \hline \end{array} \][/tex]
2. Calculate the Midpoints for Each Interval:
The midpoint of an interval is calculated by taking the average of the lower and upper bounds of the interval.
[tex]\[ \text{Midpoint} = \frac{\text{Lower bound} + \text{Upper bound}}{2} \][/tex]
- For [tex]\(120 < h \leqslant 124\)[/tex]: [tex]\(\text{Midpoint} = \frac{120+124}{2} = 122\)[/tex]
- For [tex]\(124 < h \leqslant 128\)[/tex]: [tex]\(\text{Midpoint} = \frac{124+128}{2} = 126\)[/tex]
- For [tex]\(128 < h \leqslant 132\)[/tex]: [tex]\(\text{Midpoint} = \frac{128+132}{2} = 130\)[/tex]
- For [tex]\(132 < h \leqslant 136\)[/tex]: [tex]\(\text{Midpoint} = \frac{132+136}{2} = 134\)[/tex]
- For [tex]\(136 < h \leqslant 140\)[/tex]: [tex]\(\text{Midpoint} = \frac{136+140}{2} = 138\)[/tex]
So, the midpoints are: [tex]\(122, 126, 130, 134, 138\)[/tex].
3. Calculate the Weighted Sum of the Midpoints:
Multiply each midpoint by its corresponding frequency and then sum these values.
[tex]\[ \begin{align*} \text{Weighted sum} &= (7 \times 122) + (8 \times 126) + (13 \times 130) + (9 \times 134) + (3 \times 138) \\ &= 854 + 1008 + 1690 + 1206 + 414 \\ &= 5172 \end{align*} \][/tex]
4. Calculate the Total Frequency:
Sum all the frequencies given.
[tex]\[ \text{Total frequency} = 7 + 8 + 13 + 9 + 3 = 40 \][/tex]
5. Calculate the Estimated Mean Height:
The estimated mean height is computed by dividing the weighted sum by the total frequency.
[tex]\[ \text{Mean height} = \frac{\text{Weighted sum}}{\text{Total frequency}} = \frac{5172}{40} = 129.3 \, \text{cm} \][/tex]
### Answer:
a) The estimated mean height of the students is [tex]\(129.3\)[/tex] cm.
b) The answer to part (a) is an estimate because the exact heights of the students are not given; instead, they are grouped into intervals. By using the midpoints of these intervals, we approximate the heights within each interval, leading to an estimated mean rather than an exact calculation.
### Step-by-Step Solution
1. Identify the Height Intervals and the Frequencies:
The given height intervals and their corresponding frequencies are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Height range (cm)} & \text{Frequency} \\ \hline 120 < h \leqslant 124 & 7 \\ \hline 124 < h \leqslant 128 & 8 \\ \hline 128 < h \leqslant 132 & 13 \\ \hline 132 < h \leqslant 136 & 9 \\ \hline 136 < h \leqslant 140 & 3 \\ \hline \end{array} \][/tex]
2. Calculate the Midpoints for Each Interval:
The midpoint of an interval is calculated by taking the average of the lower and upper bounds of the interval.
[tex]\[ \text{Midpoint} = \frac{\text{Lower bound} + \text{Upper bound}}{2} \][/tex]
- For [tex]\(120 < h \leqslant 124\)[/tex]: [tex]\(\text{Midpoint} = \frac{120+124}{2} = 122\)[/tex]
- For [tex]\(124 < h \leqslant 128\)[/tex]: [tex]\(\text{Midpoint} = \frac{124+128}{2} = 126\)[/tex]
- For [tex]\(128 < h \leqslant 132\)[/tex]: [tex]\(\text{Midpoint} = \frac{128+132}{2} = 130\)[/tex]
- For [tex]\(132 < h \leqslant 136\)[/tex]: [tex]\(\text{Midpoint} = \frac{132+136}{2} = 134\)[/tex]
- For [tex]\(136 < h \leqslant 140\)[/tex]: [tex]\(\text{Midpoint} = \frac{136+140}{2} = 138\)[/tex]
So, the midpoints are: [tex]\(122, 126, 130, 134, 138\)[/tex].
3. Calculate the Weighted Sum of the Midpoints:
Multiply each midpoint by its corresponding frequency and then sum these values.
[tex]\[ \begin{align*} \text{Weighted sum} &= (7 \times 122) + (8 \times 126) + (13 \times 130) + (9 \times 134) + (3 \times 138) \\ &= 854 + 1008 + 1690 + 1206 + 414 \\ &= 5172 \end{align*} \][/tex]
4. Calculate the Total Frequency:
Sum all the frequencies given.
[tex]\[ \text{Total frequency} = 7 + 8 + 13 + 9 + 3 = 40 \][/tex]
5. Calculate the Estimated Mean Height:
The estimated mean height is computed by dividing the weighted sum by the total frequency.
[tex]\[ \text{Mean height} = \frac{\text{Weighted sum}}{\text{Total frequency}} = \frac{5172}{40} = 129.3 \, \text{cm} \][/tex]
### Answer:
a) The estimated mean height of the students is [tex]\(129.3\)[/tex] cm.
b) The answer to part (a) is an estimate because the exact heights of the students are not given; instead, they are grouped into intervals. By using the midpoints of these intervals, we approximate the heights within each interval, leading to an estimated mean rather than an exact calculation.
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