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Use the histogram to complete the table below

Mathematics

ScanSolve AI Answer

Step-by-step solution

8 steps

Part a· 4 steps

Use it to fill the table below

  1. Determine the class intervals and class widths from the histogram.

    The x-axis shows the class boundaries.
    The class width is the difference between the upper and lower class boundaries.

  2. Read the frequency density for each class from the y-axis.

    Frequency density is given by the height of each bar.

  3. Calculate the frequency for each class.

    Frequency = Frequency density ×\times Class width.

  4. Calculate the midpoint for each class.

    Midpoint = (Lower class boundary + Upper class boundary) / 2.

    ClassesMidpoint (xx)Frequency DensityClass WidthFrequency (ff)
    19.5-29.524.50.4100.4×10=40.4 \times 10 = 4
    29.5-34.532.01.251.2×5=61.2 \times 5 = 6
    34.5-54.544.51.1201.1×20=221.1 \times 20 = 22
    54.5-74.564.51.3201.3×20=261.3 \times 20 = 26
    74.5-84.579.50.6100.6×10=60.6 \times 10 = 6
    84.5-94.589.50.2100.2×10=20.2 \times 10 = 2

    The completed table is:

    ClassesMidpointFrequency19.5−29.524.5429.5−34.532.0634.5−54.544.52254.5−74.564.52674.5−84.579.5684.5−94.589.52\boxed{ \begin{array}{|l|l|l|} \hline \text{Classes} & \text{Midpoint} & \text{Frequency} \\ \hline 19.5-29.5 & 24.5 & 4 \\ \hline 29.5-34.5 & 32.0 & 6 \\ \hline 34.5-54.5 & 44.5 & 22 \\ \hline 54.5-74.5 & 64.5 & 26 \\ \hline 74.5-84.5 & 79.5 & 6 \\ \hline 84.5-94.5 & 89.5 & 2 \\ \hline \end{array} }

Part b· 3 steps

Calculate the median

  1. Calculate the total frequency (NN) and the median position.

    Total frequency N=4+6+22+26+6+2=66N = 4 + 6 + 22 + 26 + 6 + 2 = 66.
    Median position = N2=662=33rd\frac{N}{2} = \frac{66}{2} = 33^{rd} value.

  2. Determine the median class.

    We need to find the class where the 33rd33^{rd} value lies.
    Cumulative frequencies:
    19.5-29.5: 4
    29.5-34.5: 4+6=104+6=10
    34.5-54.5: 10+22=3210+22=32
    54.5-74.5: 32+26=5832+26=58
    The 33rd33^{rd} value falls into the class 54.5−74.554.5 - 74.5. This is the median class.

  3. Apply the median formula for grouped data.

    The median formula is:

    Median=L+(N2−CFbfm)c\text{Median} = L + \left(\frac{\frac{N}{2} - CF_b}{f_m}\right)c

    Where:
    LL = Lower class boundary of the median class = 54.5
    NN = Total frequency = 66
    CFbCF_b = Cumulative frequency of the class before the median class = 32
    fmf_m = Frequency of the median class = 26
    cc = Class width of the median class = 74.5−54.5=2074.5 - 54.5 = 20

    Substitute the values into the formula:

    Median=54.5+(662−3226)×20\text{Median} = 54.5 + \left(\frac{\frac{66}{2} - 32}{26}\right) \times 20 Median=54.5+(33−3226)×20\text{Median} = 54.5 + \left(\frac{33 - 32}{26}\right) \times 20 Median=54.5+(126)×20\text{Median} = 54.5 + \left(\frac{1}{26}\right) \times 20 Median=54.5+2026\text{Median} = 54.5 + \frac{20}{26} Median=54.5+1013\text{Median} = 54.5 + \frac{10}{13} Median≈54.5+0.76923\text{Median} \approx 54.5 + 0.76923 Median≈55.26923\text{Median} \approx 55.26923

    Rounding to two decimal places:
    The median is 55.27\boxed{55.27}.

Part c· 1 step

Calculate the mean

  1. Calculate the sum of fxfx (frequency ×\times midpoint) for all classes.

    ClassesMidpoint (xx)Frequency (ff)fxfx
    19.5-29.524.5424.5×4=98.024.5 \times 4 = 98.0
    29.5-34.532.0632.0×6=192.032.0 \times 6 = 192.0
    34.5-54.544.52244.5×22=979.044.5 \times 22 = 979.0
    54.5-74.564.52664.5×26=1677.064.5 \times 26 = 1677.0
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