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  • Perplex
    IB Math AIHL
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    Descriptive Statistics
    /

    Problems

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    Problem Bank - Descriptive Statistics

    Access custom-built, exam-style problems for descriptive statistics. Each problem has a full solution and mark-scheme, as well as AI grading and support.

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    IB: 4
    1

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    The government of Mongolia surveys 6 herding families living in the valley of Yolyn to find out how many goats and sheep each family owns.

    Goats (g)

    17

    33

    46

    79

    93

    114

    Sheep (s)

    21

    29

    63

    85

    101

    106

    1. Find and interpret Pearson's product-moment correlation coefficient, r, for this data.

      [2]
    2. Find the equation of the regression line of s on g.

      [2]
    3. Use the regression line found in (b) to predict the number of goats (g) owned by a family in the valley with 205 sheep (s).

      [2]
    4. State two reasons why this prediction may not be accurate.

      [2]
    2

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    Dr. Darwin is interested in studying turtle hatching on a chain of 3 islands, Sandy, Blue and Eagle. Dr. Darwin first wants to estimate the total number of nests on the island chain. To do this, he first finds the total surface area of the islands, 22km2. He lands his boat on the sheltered side of the largest island, and draws a 100m by 100m square in the sand starting from the point he came ashore. He then counts the number of nests inside the square, and finds 8.

    1. State

      1. the type of variable being recorded,

        [1]
      2. the name of this type of sampling technique.

        [1]
    2. Give two reasons Darwin's measurement is unlikely to produce a reliable estimate of the total number of nests.

      [2]

    Dr. Darwin makes a second attempt at data collection. This time, he visits each island, and takes 4 separate 100m×100m samples, one on each side (north, west, south & east) of the island. The data for the number of nests found in each sample are provided in the table below.

    Island

    North

    West

    South

    East

    Sandy

    7

    4

    8

    7

    Blue

    6

    3

    7

    8

    Eagle

    3

    2

    3

    2

      1. State the name of this type of sampling technique.

        [1]
      2. Using the data collected, estimate the total number of nests across the island chain.

        [2]

    Sandy & Blue both have a surface area of 9km.

    1. Find the surface area of Eagle Island.

      [1]
      1. Explain why the estimate found in (c.iii) is unlikely to be accurate.

        [1]
      2. Using the same data as in (b.iii), give a revised estimate, to the nearest hundred, for the total number of nests across the island chain.

        [2]
    3

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    Dr. Darwin is investigating the clutch size (number of eggs per nest) of the different turtle species that nest on Eagle Island. He samples the island's nests by drawing a circle with a 100m radius centered at random GPS coordinates, and repeats this process a total of 3 times. For every nest in the sampling zones, he records the species and the number of eggs in the nest. The surface area of the island is 4km2.


    The 4 different species of marine turtles that lay eggs on the island are:

    • The Pacific Green Turtle (PG)

    • The Pacific Leatherback Turtle (LB)

    • The Indo-Pacific Hawksbill (IH)

    • The Olive Ridley Turtle (OR)


    The data collected is summarized in the table below.

    PG

    129

    130

    136

    115

    80

    106

    105

    108

    LB

    73

    102

    83

    89

    78

    105



    IH

    145

    173

    146

    158

    135

    103



    OR

    97

    103

    103

    108





    1. State the species of turtle whose clutch size has the greatest

      1. average,

        [3]
      2. variance.

        [3]
    2. Draw and label a box and whisker plot summarizing the data on Pacific Green Turtle (PG) clutch size.

      [6]
    3. Fill in the following frequency table of clutch sizes in Dr. Darwin's sample.

      [3]


    Clutch Size

    Frequency

    70-89


    90-109


    110-139


    140-159


    160-179



    1. On the grid below, sketch a cumulative frequency diagram showing the distribution of clutch sizes in Darwin's sample.

      [3]

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    4

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    The sum of the 13 elements of set S is 104.

    1. Write down the mean of S.

      [1]

    The set G is formed by doubling each element of S and adding 1.

    1. Find the mean of G.

      [1]

    It is given that the variance of G is 36.

    1. Find the standard deviation of S.

      [3]
    5

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    At the end of the 2024 fishing season on Fork Lake, 60 anglers (amateur fishers) were randomly surveyed. They were asked how many bass they caught over the whole season. The information is summarized in the following frequency table:

    Number of fish caught

    Frequency

    0

    7

    1

    12

    2

    10

    3

    19

    4

    k

    5

    3

    Assume that this sample is representative of all anglers in any season, and that all anglers have equal skill.

    1. State the value of k.

      [1]
    2. For the 2025 season estimate

      1. the probability a randomly selected angler will catch fish,

        [1]
      2. the expected number of fish caught by an angler over the season.

        [2]

    The U.S Fish and Wildlife Service (USFWS) estimates that the bass population in Lake Fork will grow by 8% between the 2024 and 2025 fishing seasons.

    1. Given that the bass population was 110′000 at the end of the 2024 season, estimate the number of bass in the Lake at the opening of the 2025 season.

      [1]

    The USFWS wants the bass population to remain unchanged year to year.

    1. How many seasonal fishing permits should be granted in 2025? Give your answer to the nearest hundred.

      [2]
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