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Statistics and Probability - Data Collection and Sampling

Grade 7IB

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

🔑Concepts

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Population and Sample: A population is the entire group you want to draw conclusions about, while a sample is a smaller group selected from that population to represent it. Visually, imagine a large container (the population) and a small scoop taking out a handful (the sample) to examine.

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Qualitative vs. Quantitative Data: Qualitative data describes qualities or categories (e.g., favorite color, brand of car), while quantitative data deals with numbers and measurements. Visually, this can be seen as a tree diagram where quantitative data branches into 'Discrete' (counted values like number of pets) and 'Continuous' (measured values like height or weight).

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Primary and Secondary Data: Primary data is information you collect yourself for a specific purpose (e.g., conducting a survey in your class). Secondary data is information collected by someone else (e.g., using data from a government website or a textbook). Visually, think of a 'Source' icon: a person with a clipboard represents primary data, while a library book or computer screen represents secondary data.

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Sampling Methods - Random Sampling: In a simple random sample, every member of the population has an equal chance of being selected. This reduces bias. Visually, this is often represented by names being pulled out of a well-shaken hat to ensure no specific group is favored.

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Bias in Sampling: Bias occurs when a sample is not representative of the population, leading to unfair or incorrect results. For example, surveying only people at a gym about their fitness habits to represent the whole city. Visually, imagine a scale that is tilted heavily to one side, showing that the data is 'weighted' unfairly.

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Frequency Tables: A frequency table is a method to organize raw data by listing each item and the number of times it occurs. Visually, the table typically includes three columns: the 'Value/Category', the 'Tally' (represented by vertical marks with a diagonal strike through every fifth mark), and the 'Frequency' (the final count written as a number).

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Discrete vs. Continuous Data: Discrete data can only take specific values (usually whole numbers), while continuous data can take any value within a range. Visually, discrete data looks like steps on a ladder (1,2,3...1, 2, 3...), whereas continuous data is like a smooth ramp where you can stop at any decimal point (1.5,1.55,1.552...1.5, 1.55, 1.552...).

📐Formulae

Mean=∑xn\text{Mean} = \frac{\sum x}{n}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Relative Frequency=Frequency of an eventTotal number of observations\text{Relative Frequency} = \frac{\text{Frequency of an event}}{\text{Total number of observations}}

Estimated Population Frequency=Sample FrequencySample Size×Population Size\text{Estimated Population Frequency} = \frac{\text{Sample Frequency}}{\text{Sample Size}} \times \text{Population Size}

💡Examples

Problem 1:

A school has 800800 students. A student wants to estimate how many students prefer chocolate milk over regular milk. They survey a random sample of 5050 students and find that 1212 prefer chocolate milk. Estimate the total number of students in the school who prefer chocolate milk.

Solution:

Step 1: Identify the sample size (n=50n = 50) and the sample frequency (1212). Step 2: Calculate the fraction of the sample that prefers chocolate milk: 1250\frac{12}{50}. Step 3: Multiply this fraction by the total population size (800800): Estimate=1250×800\text{Estimate} = \frac{12}{50} \times 800 Step 4: Simplify the calculation: Estimate=0.24×800=192\text{Estimate} = 0.24 \times 800 = 192

Explanation:

We use the proportion found in the sample to make an inference about the entire population. By assuming the sample is representative, we scale the sample results up to the size of the school population.

Problem 2:

Classify the following data sets as either Discrete or Continuous:

  1. The number of goals scored in a soccer match.
  2. The time taken to run 100100 meters.
  3. The number of students in a classroom.
  4. The temperature of a cup of tea.

Solution:

  1. Discrete (You count the goals; you cannot score 2.52.5 goals).
  2. Continuous (Time is measured and can be broken down into infinite decimals like 12.4512.45 seconds).
  3. Discrete (You count individual students; you cannot have half a student).
  4. Continuous (Temperature is measured on a scale and can be any value like 72.4∘C72.4^{\circ}C).

Explanation:

Discrete data is counted (usually whole numbers), while continuous data is measured (can include fractions and decimals).