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Statistics and Probability - Sampling

Grade 11IB_AI

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

🔑Concepts

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Population vs. Sample: A population represents the entire group under study (size NN), while a sample is a subset of the population (size nn) used to represent the whole.

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Simple Random Sampling: Every member of the population has an equal chance of being selected. This is typically achieved using a random number generator to select nn unique IDs.

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Systematic Sampling: A method where every kthk^{th} member of the population is selected. The interval is calculated as k=Nnk = \frac{N}{n}. A random starting point between 11 and kk is chosen first.

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Stratified Sampling: The population is divided into mutually exclusive groups called 'strata' based on shared characteristics (e.g., age, gender). A random sample is then taken from each stratum in proportion to the stratum's size in the population.

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Quota Sampling: Similar to stratified sampling, the population is divided into groups, but the researcher uses non-random methods (like convenience) to fill a pre-defined quota for each group.

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Convenience Sampling: Selecting subjects who are easiest to reach or most available. This method is often prone to bias as it is rarely representative of the whole population.

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Bias: Sampling bias occurs when certain members of the population are systematically more likely to be selected than others, leading to results that do not accurately reflect the population parameters.

📐Formulae

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

nstratum=NstratumNtotal×nsamplen_{stratum} = \frac{N_{stratum}}{N_{total}} \times n_{sample}

xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

💡Examples

Problem 1:

A sports club has 120120 male members and 8080 female members. A researcher wants to take a stratified sample of size n=30n = 30. Calculate the number of male and female members that should be included in the sample.

Solution:

First, calculate the total population: N=120+80=200N = 120 + 80 = 200 Next, find the number of males for the sample: nmale=120200×30=18n_{male} = \frac{120}{200} \times 30 = 18 Finally, find the number of females for the sample: nfemale=80200×30=12n_{female} = \frac{80}{200} \times 30 = 12

Explanation:

In stratified sampling, the sample size of each group is proportional to its size in the total population. Here, 60%60\% of the population is male, so 60%60\% of 3030 is 1818.

Problem 2:

A school has 15001500 students. A teacher wants to perform systematic sampling to select 6060 students for a survey. Determine the sampling interval kk and describe how the first student is chosen.

Solution:

Calculate the interval kk: k=150060=25k = \frac{1500}{60} = 25 The teacher should choose a random integer rr such that 1≤r≤251 \le r \le 25.

Explanation:

The sampling interval kk is the population size divided by the desired sample size. The first student is chosen randomly within the first interval to ensure the starting point doesn't introduce immediate bias.

Problem 3:

Identify the type of sampling used: A researcher stands outside a grocery store and asks the first 5050 people who walk out about their shopping preferences.

Solution:

Convenience Sampling.

Explanation:

The researcher is choosing participants based on ease of access and proximity rather than using a randomized process that gives every member of the population a known chance of being selected.

Sampling Grade 11 Notes & Examples | IB AI Maths