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NCERT Exemplar · Class 10 Mathematics Statistics and Probability

96 questions · 96 still being checked

EXERCISE 13.3 1–10 (part 6 of 11)

  1. Exercise 1

    Find the mean of the distribution :
    Class1\displaystyle 1-3\displaystyle 33\displaystyle 3-5\displaystyle 55\displaystyle 5-7\displaystyle 77\displaystyle 7-10\displaystyle 10
    Frequency9\displaystyle 922\displaystyle 2227\displaystyle 2717\displaystyle 17

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    NCERT’s answer
    5.$\displaystyle 5$
    Class marks and frequencies: \[x_i:\ 2,\ 4,\ 6,\ 8.5 \qquad f_i:\ 9,\ 22,\ 27,\ 17 \] \[\sum f_i = 75, \qquad \sum f_i x_i = 18+88+162+144.5 = 412.5 \] \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{412.5}{75} = 5.5 \] Answer: Mean = $\displaystyle 5.5$

  2. Exercise 2

    Calculate the mean of the scores of 20\displaystyle 20 students in a mathematics test :
    Marks10\displaystyle 10-20\displaystyle 2020\displaystyle 20-30\displaystyle 3030\displaystyle 30-40\displaystyle 4040\displaystyle 40-50\displaystyle 5050\displaystyle 50-60\displaystyle 60
    Number of students2\displaystyle 24\displaystyle 47\displaystyle 76\displaystyle 61\displaystyle 1

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    NCERT’s answer
    $\displaystyle 35$
    Class marks and frequencies: \[x_i:\ 15,\ 25,\ 35,\ 45,\ 55 \qquad f_i:\ 2,\ 4,\ 7,\ 6,\ 1 \] \[\sum f_i = 20, \qquad \sum f_i x_i = 30+100+245+270+55 = 700 \] \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{700}{20} = 35 \] Answer: Mean = $\displaystyle 35$

  3. Exercise 3

    Calculate the mean of the following data :
    Class4\displaystyle 4-7\displaystyle 78\displaystyle 8-11\displaystyle 1112\displaystyle 12-15\displaystyle 1516\displaystyle 16-19\displaystyle 19
    Frequency5\displaystyle 54\displaystyle 49\displaystyle 910\displaystyle 10

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    NCERT’s answer
    12.$\displaystyle 93$
    Class marks and frequencies: \[x_i:\ 5.5,\ 9.5,\ 13.5,\ 17.5 \qquad f_i:\ 5,\ 4,\ 9,\ 10 \] \[\sum f_i = 28, \qquad \sum f_i x_i = 27.5+38+121.5+175 = 362 \] \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{362}{28} \approx 12.93 \] Answer: Mean \(\displaystyle \approx 12.93\)

  4. Exercise 4

    The following table gives the number of pages written by Sarika for completing her own book for 30\displaystyle 30 days :
    Number of pages written per day16\displaystyle 16-18\displaystyle 1819\displaystyle 19-21\displaystyle 2122\displaystyle 22-24\displaystyle 2425\displaystyle 25-27\displaystyle 2728\displaystyle 28-30\displaystyle 30
    Number of days1\displaystyle 13\displaystyle 34\displaystyle 49\displaystyle 913\displaystyle 13
    Find the mean number of pages written per day.

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    NCERT’s answer
    $\displaystyle 26$
    Class marks and frequencies: \[x_i:\ 17,\ 20,\ 23,\ 26,\ 29 \qquad f_i:\ 1,\ 3,\ 4,\ 9,\ 13 \] \[\sum f_i = 30, \qquad \sum f_i x_i = 17+60+92+234+377 = 780 \] \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{780}{30} = 26 \] Answer: $\displaystyle 26$ pages per day

  5. Exercise 5

    The daily income of a sample of 50\displaystyle 50 employees are tabulated as follows :
    Income (in Rs)1\displaystyle 1-200\displaystyle 200201\displaystyle 201-400\displaystyle 400401\displaystyle 401-600\displaystyle 600601\displaystyle 601-800\displaystyle 800
    Number of employees14\displaystyle 1415\displaystyle 1514\displaystyle 147\displaystyle 7
    Find the mean daily income of employees.

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    NCERT’s answer
    Rs. $\displaystyle 356.5$
    Class marks and frequencies: \[x_i:\ 100.5,\ 300.5,\ 500.5,\ 700.5 \qquad f_i:\ 14,\ 15,\ 14,\ 7 \] \[\sum f_i = 50, \qquad \sum f_i x_i = 1407+4507.5+7007+4903.5 = 17825 \] \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{17825}{50} = 356.5 \] Answer: Rs $\displaystyle 356.5$

  6. Exercise 6

    An aircraft has 120\displaystyle 120 passenger seats. The number of seats occupied during 100\displaystyle 100 flights is given in the following table :
    Number of seats100\displaystyle 100-104\displaystyle 104104\displaystyle 104-108\displaystyle 108108\displaystyle 108-112\displaystyle 112112\displaystyle 112-116\displaystyle 116116\displaystyle 116-120\displaystyle 120
    Frequency15\displaystyle 1520\displaystyle 2032\displaystyle 3218\displaystyle 1815\displaystyle 15
    Determine the mean number of seats occupied over the flights.

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    NCERT’s answer
    109. $\displaystyle 92$
    Class marks, assumed mean \(\displaystyle A=110\), \(\displaystyle h=4\): \[x_i:\ 102,\ 106,\ 110,\ 114,\ 118 \qquad f_i:\ 15,\ 20,\ 32,\ 18,\ 15 \] \[u_i = \frac{x_i-A}{h}:\ -2,\ -1,\ 0,\ 1,\ 2 \] \[\sum f_i = 100, \qquad \sum f_i u_i = -30-20+0+18+30 = -2 \] \[\bar{x} = A + h\!\left(\frac{\sum f_i u_i}{\sum f_i}\right) = 110 + 4\!\left(\frac{-2}{100}\right) = 109.92 \] Answer: $\displaystyle 109.92$ seats

  7. Exercise 7

    The weights (in kg) of 50\displaystyle 50 wrestlers are recorded in the following table :
    Weight (in kg)100\displaystyle 100-110\displaystyle 110110\displaystyle 110-120\displaystyle 120120\displaystyle 120-130\displaystyle 130130\displaystyle 130-140\displaystyle 140140\displaystyle 140-150\displaystyle 150
    Number of wrestlers4\displaystyle 414\displaystyle 1421\displaystyle 218\displaystyle 83\displaystyle 3
    Find the mean weight of the wrestlers.

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    NCERT’s answer
    123.$\displaystyle 4$ kg
    Class marks, assumed mean \(\displaystyle A=125\), \(\displaystyle h=10\): \[x_i:\ 105,\ 115,\ 125,\ 135,\ 145 \qquad f_i:\ 4,\ 14,\ 21,\ 8,\ 3 \] \[u_i = \frac{x_i-A}{h}:\ -2,\ -1,\ 0,\ 1,\ 2 \] \[\sum f_i = 50, \qquad \sum f_i u_i = -8-14+0+8+6 = -8 \] \[\bar{x} = A + h\!\left(\frac{\sum f_i u_i}{\sum f_i}\right) = 125 + 10\!\left(\frac{-8}{50}\right) = 123.4 \] Answer: $\displaystyle 123.4$ kg

  8. Exercise 8

    The mileage (km per litre) of 50\displaystyle 50 cars of the same model was tested by a manufacturer and details are tabulated as given below :
    Mileage (km/l)10\displaystyle 10-12\displaystyle 1212\displaystyle 12-14\displaystyle 1414\displaystyle 14-16\displaystyle 1616\displaystyle 16-18\displaystyle 18
    Number of cars7\displaystyle 712\displaystyle 1218\displaystyle 1813\displaystyle 13
    Find the mean mileage. The manufacturer claimed that the mileage of the model was 16\displaystyle 16 km/litre. Do you agree with this claim?

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    NCERT’s answer
    14.$\displaystyle 48$ km/l; No, the manufacturer is claiming mileage $\displaystyle 1.52$ km/h more than the average mileage
    \[\begin{array}{|c|c|c|c|}\hline x_i & f_i & u_i=\dfrac{x_i-15}{2} & f_iu_i \\ \hline 11 & 7 & -2 & -14 \\ 13 & 12 & -1 & -12 \\ 15 & 18 & 0 & 0 \\ 17 & 13 & 1 & 13 \\ \hline & 50 & & -13 \\ \hline \end{array} \] \[\bar{x} = a + h\cdot\dfrac{\sum f_iu_i}{\sum f_i} \] \[\bar{x} = 15 + 2\left(\dfrac{-13}{50}\right) \] \[\bar{x} = 15 - 0.52 = 14.48 \] \(\displaystyle 14.48 < 16\), so the sample does not support the manufacturer's claim.Answer: Mean mileage \(\displaystyle =14.48\) km/l; the claim of $\displaystyle 16$ km/l is not supported.

  9. Exercise 9

    The following is the distribution of weights (in kg) of 40\displaystyle 40 persons :
    Weight (in kg)40\displaystyle 40-45\displaystyle 4545\displaystyle 45-50\displaystyle 5050\displaystyle 50-55\displaystyle 5555\displaystyle 55-60\displaystyle 6060\displaystyle 60-65\displaystyle 6565\displaystyle 65-70\displaystyle 7070\displaystyle 70-75\displaystyle 7575\displaystyle 75-80\displaystyle 80
    Number of persons4\displaystyle 44\displaystyle 413\displaystyle 135\displaystyle 56\displaystyle 65\displaystyle 52\displaystyle 21\displaystyle 1
    Construct a cumulative frequency distribution (of the less than type) table for the data above.

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    NCERT’s answer
    Weight (in kg)Number of persons
    Less then $\displaystyle 45$$\displaystyle 4$
    Less then $\displaystyle 50$$\displaystyle 8$
    Less then $\displaystyle 55$$\displaystyle 21$
    Less then $\displaystyle 60$$\displaystyle 26$
    Less then $\displaystyle 65$$\displaystyle 32$
    Less then $\displaystyle 70$$\displaystyle 37$
    Less then $\displaystyle 75$$\displaystyle 39$
    Less then $\displaystyle 80$$\displaystyle 40$
    Running totals of the frequencies, taken up to each class's upper limit, give the less-than-type cumulative frequencies. \[\begin{array}{|l|c|}\hline \text{Weight (in kg)} & \text{Cumulative frequency} \\ \hline \text{Less than }45 & 4 \\ \text{Less than }50 & 8 \\ \text{Less than }55 & 21 \\ \text{Less than }60 & 26 \\ \text{Less than }65 & 32 \\ \text{Less than }70 & 37 \\ \text{Less than }75 & 39 \\ \text{Less than }80 & 40 \\ \hline \end{array} \]Answer: cumulative frequencies $\displaystyle 4$, $\displaystyle 8$, $\displaystyle 21$, $\displaystyle 26$, $\displaystyle 32$, $\displaystyle 37$, $\displaystyle 39$, $\displaystyle 40$ (less than type).

  10. Exercise 10

    The following table shows the cumulative frequency distribution of marks of 800\displaystyle 800 students in an examination:
    MarksNumber of students
    Below 10\displaystyle 1010\displaystyle 10
    Below 20\displaystyle 2050\displaystyle 50
    Below 30\displaystyle 30130\displaystyle 130
    Below 40\displaystyle 40270\displaystyle 270
    Below 50\displaystyle 50440\displaystyle 440
    Below 60\displaystyle 60570\displaystyle 570
    Below 70\displaystyle 70670\displaystyle 670
    Below 80\displaystyle 80740\displaystyle 740
    Below 90\displaystyle 90780\displaystyle 780
    Below 100\displaystyle 100800\displaystyle 800
    Construct a frequency distribution table for the data above.

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    NCERT’s answer
    MarksNumber of students
    $\displaystyle 0$-$\displaystyle 10$$\displaystyle 10$
    $\displaystyle 10$-$\displaystyle 20$$\displaystyle 40$
    $\displaystyle 20$-$\displaystyle 30$$\displaystyle 80$
    $\displaystyle 30$-$\displaystyle 40$$\displaystyle 140$
    $\displaystyle 40$-$\displaystyle 50$$\displaystyle 170$
    $\displaystyle 50$-$\displaystyle 60$$\displaystyle 130$
    $\displaystyle 60$-$\displaystyle 70$$\displaystyle 100$
    $\displaystyle 70$-$\displaystyle 80$$\displaystyle 70$
    $\displaystyle 80$-$\displaystyle 90$$\displaystyle 40$
    $\displaystyle 90$-$\displaystyle 100$$\displaystyle 20$
    Each class frequency is the difference of consecutive below-type cumulative frequencies. \[\begin{array}{|c|c|}\hline \text{Marks} & \text{Number of students} \\ \hline 0\text{-}10 & 10 \\ 10\text{-}20 & 40 \\ 20\text{-}30 & 80 \\ 30\text{-}40 & 140 \\ 40\text{-}50 & 170 \\ 50\text{-}60 & 130 \\ 60\text{-}70 & 100 \\ 70\text{-}80 & 70 \\ 80\text{-}90 & 40 \\ 90\text{-}100 & 20 \\ \hline \text{Total} & 800 \\ \hline \end{array} \]Answer: frequencies $\displaystyle 10$, $\displaystyle 40$, $\displaystyle 80$, $\displaystyle 140$, $\displaystyle 170$, $\displaystyle 130$, $\displaystyle 100$, $\displaystyle 70$, $\displaystyle 40$, $\displaystyle 20$ (total $\displaystyle 800$).