NeokaalStudy

Statistics: Competency Practice

Expressway radar speed analysis, dual missing frequency median solver, step-deviation mean, and cumulative conversions.

6 problems·20–25 min·★★★★★
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  1. Problem 1
    statistics·★★★★★
    Speed radar recordings of $140$ vehicles on an expressway are grouped into speed ranges (km/h) with frequencies: $40-60: 14$, $60-80: 30$, $80-100: 50$, $100-120: 32$, $120-140: 14$. (i) Identify the modal class of the speed distribution. (ii) Calculate the modal speed of the vehicles. (iii) If the speed limit is $100\text{ km/h}$, find the percentage of vehicles that exceeded the speed limit.
    ▶Answer
    (i) Modal class: $80 - 100\text{ km/h}$, (ii) Mode $\approx 90.53\text{ km/h}$, (iii) $32.9\%$
    ▶Step-by-step solution
    1. (i) The highest frequency is $f_1 = 50$ corresponding to the interval $80 - 100\text{ km/h}$, so the modal class is $80 - 100\text{ km/h}$.
    2. (ii) Using $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)h = 80 + \left(\frac{20}{38}\right)20 \approx 90.53\text{ km/h}$.
    3. (iii) Vehicles exceeding $100\text{ km/h}$ are in classes $100-120$ and $120-140$ ($46$ vehicles), giving $\frac{46}{140} \times 100\% = 32.9\%$.
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  2. Problem 2
    grouped-median·★★★★★
    The median of classes 0-10, 10-20, 20-30, 30-40, 40-50, 50-60 with frequencies $5,x,20,15,y,5$ is 28.5 and the total frequency is 60. Find $x$ and $y$.
    ▶Answer
    $x=8,\ y=7$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
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  3. Problem 3
    grouped-mean·★★★★☆
    Equal-width classes have midpoints [25, 35, 45, 55] and frequencies [5, 2, 6, 3]. Find the mean using step-deviation.
    ▶Answer
    $\frac{315}{8} \approx 39.38$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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  4. Problem 4
    grouped-median·★★★★☆
    A less-than cumulative-frequency table at upper boundaries 10, 20, 30, 40 is [6, 11, 15, 21]. Convert it to class frequencies and find the median.
    ▶Answer
    $19.00$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
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  5. Problem 5
    grouped-mode·★★★☆☆
    The modal class is 10-20, with preceding, modal, and succeeding frequencies 7, 10, 8. Find the mode.
    ▶Answer
    $16.00$
    ▶Step-by-step solution
    1. Identify the modal class and its neighbouring frequencies.
    2. Use the grouped-mode formula, or $\text{Mode}\approx3\text{Median}-2\text{Mean}$ when requested.
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  6. Problem 6
    grouped-mean·★★★★☆
    A grouped distribution has midpoints [12, 14, 16, 18, 20, 22, 24] and frequencies $[7,6,9,13,f,5,4]$. Its mean is 18. Find $f$.
    ▶Answer
    $20$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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