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Given the heights of students in an image, construct a cumulative frequency table using 10cm intervals starting from 120cm.

Mathematics

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Answer

b

7 students\text{7 students}

Step-by-step solution

5 steps

First, I will list the heights from the image. I can identify 32 heights from the provided image. Although the question states "42 students", I will proceed with the 32 visible data points and note this assumption.

The heights are:
155, 134, 164, 174, 126, 158, 168, 162, 142, 154, 159, 179, 137, 157, 166, 151, 136, 145, 173, 168, 157, 163, 165, 152, 140, 169, 154, 167, 171, 157, 160, 158.

Sorting these heights in ascending order:
126, 134, 136, 137, 140, 142, 145, 151, 152, 154, 154, 155, 157, 157, 157, 158, 158, 159, 160, 162, 163, 164, 165, 166, 167, 168, 168, 169, 171, 173, 174, 179.

Part a· 3 steps

Grouping the height in intervals of 10cm, starting at 120cm. Construct the Cumulative Frequency Table for the data.

  1. Define the class intervals.

    The intervals are 10cm wide, starting at 120cm.

    120≤height<130130≤height<140140≤height<150150≤height<160160≤height<170170≤height<180\begin{align*} \text{120} & \le \text{height} < \text{130} \\ \text{130} & \le \text{height} < \text{140} \\ \text{140} & \le \text{height} < \text{150} \\ \text{150} & \le \text{height} < \text{160} \\ \text{160} & \le \text{height} < \text{170} \\ \text{170} & \le \text{height} < \text{180} \end{align*}

  2. Count the frequency for each interval.

    • 120 - 129 cm: 126 (1 student)
    • 130 - 139 cm: 134, 136, 137 (3 students)
    • 140 - 149 cm: 140, 142, 145 (3 students)
    • 150 - 159 cm: 151, 152, 154, 154, 155, 157, 157, 157, 158, 158, 159 (11 students)
    • 160 - 169 cm: 160, 162, 163, 164, 165, 166, 167, 168, 168, 169 (10 students)
    • 170 - 179 cm: 171, 173, 174, 179 (4 students)

  3. Construct the Cumulative Frequency Table.

    \begin{tabular}{|c|c|c|}
    \hline
    Height (cm) & Frequency (f) & Cumulative Frequency (cf) \ \hline
    120 - 129 & 1 & 1 \ 130 - 139 & 3 & 1+3=41 + 3 = 4 \ 140 - 149 & 3 & 4+3=74 + 3 = 7 \ 150 - 159 & 11 & 7+11=187 + 11 = 18 \ 160 - 169 & 10 & 18+10=2818 + 10 = 28 \ 170 - 179 & 4 & 28+4=3228 + 4 = 32 \ \hline
    \end{tabular}

Part b

7 students\text{7 students}

Students who are shorter than 150cm pay half price to enter a team park. How many of these students are there?

  1. Identify the heights shorter than 150cm.

    This includes all students in the height intervals below 150cm.
    These are the students in the 120-129 cm, 130-139 cm, and 140-149 cm intervals.

  2. Sum the frequencies for these intervals.

    Number of students = (Frequency for 120-129 cm) + (Frequency for 130-139 cm) + (Frequency for 140-149 cm)
    Number of students = 1+3+3=71 + 3 + 3 = 7

    Alternatively, from the cumulative frequency table, the cumulative frequency for the "140 - 149" class is 7, which represents all students with heights less than 150 cm.

    There are 7 students\boxed{\text{7 students}} who are shorter than 150cm.

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