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Statistics and Probability - Data collection

Grade 12IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A population is the entire group that is being studied, while a sample is a sub-collection of members selected from that population.

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Data can be Qualitative (categorical/descriptive) or Quantitative (numerical).

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Quantitative data is classified as Discrete (countable values, such as the number of students n∈{0,1,2,...}n \in \{0, 1, 2, ...\}) or Continuous (measurable values within a range, such as height 150<h<190150 < h < 190 cm).

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Simple Random Sampling: Every member of the population has an equal chance of being selected, often using a random number generator.

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Systematic Sampling: Selecting every kthk^{th} member from a list of the population.

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Stratified Sampling: The population is divided into mutually exclusive groups (strata), and a random sample is taken from each group proportional to the size of the group in the population.

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Quota Sampling: Similar to stratified sampling, but the selection within groups is non-random (e.g., convenience).

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Convenience Sampling: Selecting subjects that are easiest to reach or most available.

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Sampling Bias occurs when certain members of the population are more or less likely to be included than others, leading to a non-representative sample.

📐Formulae

k=Nnk = \frac{N}{n}

Sample Size of Stratum=Stratum SizeTotal Population Size×Total Sample Size\text{Sample Size of Stratum} = \frac{\text{Stratum Size}}{\text{Total Population Size}} \times \text{Total Sample Size}

💡Examples

Problem 1:

A high school has 1200 students. A researcher wants to conduct a systematic sample of 60 students. Determine the sampling interval kk.

Solution:

Using the formula k=Nnk = \frac{N}{n}: k=120060k = \frac{1200}{60} k=20k = 20

Explanation:

The researcher should select every 20th20^{th} student from the school registry to obtain a sample of 60.

Problem 2:

A sports club has 200 junior members and 300 senior members. A researcher wants a stratified sample of 50 members. How many junior members should be in the sample?

Solution:

Total population N=200+300=500N = 200 + 300 = 500. Junior Sample=200500×50\text{Junior Sample} = \frac{200}{500} \times 50 Junior Sample=0.4×50\text{Junior Sample} = 0.4 \times 50 Junior Sample=20\text{Junior Sample} = 20

Explanation:

To maintain the same proportion as the population, 20 junior members must be selected.

Problem 3:

Identify the type of data for the following: i) Number of goals scored in a match (gg), ii) Time taken to run 100m (tt).

Solution:

i) gg is Quantitative Discrete. ii) tt is Quantitative Continuous.

Explanation:

Goals are counted in whole numbers (you cannot score 2.52.5 goals), making it discrete. Time is measured and can take any value on a scale, making it continuous.