Sampling and Statistics Chapter Notes - Grade 7 PDF Download

Table of Contents
1. Introduction
2. Biased and Unbiased Samples
3. Make Predictions
4. Generate Multiple Samples
5. Compare Two Populations
6. Assess Visual Overlap
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Introduction

Biased and Unbiased Samples

What is a sample?

• A sample is a small part of a larger group (called the population) we study to learn about the whole group.
• It saves time since we don't need to ask everyone in the population.
• Example: To find out what games 1000 students in a school like, we might survey 50 students.

Unbiased Sample

• An unbiased sample fairly represents the population.
• Every person in the population has an equal chance of being chosen.
  • Example: Randomly picking 20 students from a list of all 500 students in a school.
• Unbiased samples help us make accurate conclusions.

Biased Sample

• A biased sample does not fairly represent the population.
• Some people are more likely to be chosen than others.
  • Example: Asking only students in the chess club about favorite games is biased because it ignores students who don't play chess.
• Biased samples can lead to wrong conclusions about the population.

For Example: A school has 200 students: You want to know their favorite subject. You survey 10 students from the math club. Is this sample biased or unbiased?
Solution: The sample is biased because only math club students are surveyed, and they may prefer math more than other students.

Make Predictions

What is a prediction?

• A prediction is an educated guess about the population based on sample data.
• It uses patterns in the sample to estimate what the whole group is like.

Using samples to predict

• Take an unbiased sample from the population.
• Analyze the sample data to find patterns, like percentages or averages.
• Use the patterns to predict the population's behavior.

For Example: In a sample of 20 students, 12 say they like pizza. Predict how many students in a school of 400 like pizza.
Solution:

• Find the proportion in the sample: 12 ÷ 20 = 0.6 or 60%.
• Apply the proportion to the population: 0.6 × 400 = 240.
• Prediction: About 240 students like pizza.

Why unbiased samples matter

• Unbiased samples give more reliable predictions.
• Biased samples can lead to incorrect predictions, like overestimating or underestimating.

Generate Multiple Samples

What are multiple samples?

• Multiple samples are several different samples taken from the same population.
• Each sample is chosen randomly and independently.

Why use multiple samples?

• They help check if results are consistent across different samples.
• They give a clearer picture of the population.

Explore: Multiple Samples

• Take several random samples from the population.
• Compare the results to see if they are similar.
  • Example: Survey three groups of 10 students each about favorite ice cream flavors.
• If all three samples show about 50% like chocolate, the result is reliable.

Explore: Sample Size in Multiple Samples

• Sample size is the number of items or people in a sample.
• Larger samples tend to give results closer to the true population.
• Small samples may vary a lot and be less reliable.
  • Example Problem: You survey two samples about favorite sports. Sample A has 5 students, and 3 like soccer (60%). Sample B has 50 students, and 25 like soccer (50%). Which is more reliable?
  • Solution: Sample B is more reliable because its larger size (50) gives a better representation of the population.

For Example: Three samples of sizes 10, 20, and 30 students show 40%, 45%, and 43% prefer reading. Find the average percentage.
Solution:

• Add the percentages: 40 + 45 + 43 = 128.
• Divide by the number of samples: 128 ÷ 3 ≈ 42.67.
• Average: About 42.67% prefer reading.

Compare Two Populations

What does it mean to compare populations?

• Comparing populations means looking at two groups to see how they differ or are similar.
• Example: Comparing the heights of 7th graders in two different schools.

Using samples to compare

• Take random samples from each population.
• Collect data, like heights or test scores.
• Compare the data using averages or other measures.

Explore: Compare Means of Two Populations

• The mean is the average, found by adding all numbers and dividing by the count.
  • Formula: Mean = (Sum of values) ÷ (Number of values).
• Compare the means of two samples to see if the populations differ.

For Example: Sample A from School 1 has test scores {80, 85, 90} (3 students). Sample B from School 2 has test scores {75, 80, 85, 90} (4 students). Compare the means.
Solution:

• School 1 mean: (80 + 85 + 90) ÷ 3 = 255 ÷ 3 = 85.
• School 2 mean: (75 + 80 + 85 + 90) ÷ 4 = 330 ÷ 4 = 82.5.
• Comparison: ​School 1's mean (85) is higher than School 2's (82.5), suggesting School 1's students may score slightly higher.

Assess Visual Overlap

What is visual overlap?

• Visual overlap happens when data from two populations looks similar on a graph.
• Graphs like dot plots or histograms show how data is spread.

How to assess visual overlap?

• Create a graph (like a dot plot) for each population's sample.
• Compare the graphs to see if they overlap.
• Lots of overlap means the populations are similar; little overlap means they are different.

Example Problem

• Two classes' test scores are: Class A {70, 75, 80, 85} and Class B {80, 85, 90, 95}. You make dot plots and notice the plots barely overlap. What does this mean?
• Solution: Little overlap suggests Class B's scores are generally higher than Class A's.

Types of graphs

• Dot plots: Show each data point as a dot on a number line.
• Histograms: Group data into ranges and show frequency.
• Box plots: Show the range, middle, and spread of data.

For Example: Calculate the range for Class A's scores {70, 75, 80, 85} to understand spread for a box plot.
Solution:

• Range = Highest value - Lowest value.
• Range = 85 - 70 = 15.
• The range (15) helps show how spread out Class A's scores are.