Draw Inferences From
Random Samples
Learn how to collect random samples, calculate statistics, and use them to make predictions about entire populations.
What Is a Population?
A population is the entire group you want to learn about. It could be every student in your school, every tree in a forest, or every word in a book.
What Is a Random Sample?
A random sample is a subset of the population chosen so that every member has an equal chance of being selected. This prevents bias and makes the sample representative.
Why Does Random Matter?
If you only survey your friends about a school election, the results may not reflect the whole school. A random sample avoids this bias and gives every voice an equal chance.
The Core Inference Principle
The sample mean \(\bar{x}\) approximates the population mean \(\mu\). The larger and more random the sample, the closer the estimate.
Taking Multiple Samples
One sample gives one estimate. Taking multiple samples of the same size lets you see how much the estimates vary, which tells you how reliable your prediction is.
Sample vs. Census vs. Inference
How to Draw an Inference
Example 1: Mean Word Length
A student randomly selects 10 words from a book and records their lengths (in letters):
Step 1: Add the values: \(3 + 5 + 7 + 4 + 6 + 2 + 8 + 5 + 4 + 6 = 50\)
Step 2: Divide by the sample size: \(\bar{x} = \frac{50}{10} = 5\) letters
Example 2: School Election
There are 800 students in a school. A random sample of 50 students is surveyed about their vote for class president:
Example 3: Gauging Variation
A student takes four random samples of 10 words each from a book and finds these sample means:
| Sample | Mean (letters) |
|---|---|
| Sample 1 | 5.2 |
| Sample 2 | 4.8 |
| Sample 3 | 5.5 |
| Sample 4 | 4.9 |
Range of means: \(5.5 - 4.8 = 0.7\) letters. The estimates cluster between 4.8 and 5.5, so the true population mean is likely in that range.
Bigger Samples, Less Variation
Using Proportions to Predict
Sometimes you want to estimate what fraction of a population has a certain trait. The sample proportion is your estimate.
Example: In a random sample of 40 students, 12 prefer pizza for lunch.
\(\hat{p} = \frac{12}{40} = 0.30\), so we estimate about 30% of all students prefer pizza.
If the school has 600 students: \(0.30 \times 600 = \) 180 students likely prefer pizza.
Common Mistakes
Summary Comparison
| Population | Sample | |
|---|---|---|
| Definition | Entire group of interest | Subset randomly chosen |
| Size | Often very large | Manageable (e.g., 30-50) |
| Mean symbol | \(\mu\) | \(\bar{x}\) |
| Proportion symbol | \(p\) | \(\hat{p}\) |
| Known? | Usually unknown | Calculated from data |
| Purpose | What we want to learn | How we estimate it |
Question 1 of 5
A teacher wants to estimate the average number of books read by students in a school of 500. She randomly surveys 40 students and finds a sample mean of 12 books. What is the best inference?
Question 2 of 5
Which of the following is a random sample of students in a school?
Question 3 of 5
A student takes three random samples of 20 words from a book and finds sample means of 4.8, 5.1, and 5.0 letters. What is the best estimate for the population mean word length?
Question 4 of 5
In a random sample of 60 students, 24 say they walk to school. If the school has 750 students total, about how many students likely walk to school?
Question 5 of 5
Why does taking multiple random samples improve your inference about a population?