Chapter 6: Sample Surveys
Hello everyone! We are finally in the home stretch of our math studies.
The topic we are covering today, "Sample Surveys," is incredibly practical and used all around us in our daily lives.
For example, it's the logic behind "TV ratings," "election exit polls," and "product quality inspections." We will learn the wisdom of how to investigate things when it’s difficult or impossible to check everything!
The feeling that "it seems like a huge hassle to check everything..." is actually the first step toward understanding this unit. So, relax and let's jump in!
1. Check Everything or Just a Part?
There are two main methods for conducting surveys. Let’s make sure we clearly understand the difference between them.
① Census (Complete Enumeration)
A survey that examines everything in the target group.
(Examples: National census, school health checkups, grading every test, etc.)
We use this when accurate data is required, but it costs a lot of time and money.
② Sample Survey
A survey that takes a small part of the target group and estimates the overall state from that information.
(Examples: TV ratings, light bulb lifespan testing, checking water quality in a river, etc.)
If you were to do a "census" on the lifespan of light bulbs, you’d have to use them all up and have nothing left to sell, right? That’s why we need sample surveys to check just a portion.
【Key Point】
Use a sample survey for things that must be destroyed to be tested (like lifespan) or cases where there are so many items that a full inspection is impossible.
2. Learn the Essential Keywords!
There are a few new terms in sample surveys. It’s easier to remember if we compare it to cooking!
・Population (母集団): The entire group that is the target of the survey. (Example: A whole pot of soup.)
・Sample (標本): A portion taken from the population. (Example: A spoonful taken in a small bowl to taste-test.)
・Extraction/Sampling (抽出): Taking a sample from the population. (Example: Scooping soup with a ladle.)
・Sample Size (標本の大きさ): The number of items or data points in the sample. (Example: Whether you took one spoon or one whole bowl.)
【Pro Tip】
Be careful not to confuse "Sample Size" with "Number of Samples"!
If you choose 100 people, the size is "100." It does not refer to how many groups you divided them into (like "Group 1, Group 2...").
3. The Secret to Accuracy: "Randomness"
The most important thing in a sample survey is to choose without "bias."
For example, if you wanted to know what junior high students like to eat, but you only asked students in the sports clubs, you might just get "Meat!" as the answer. That wouldn't represent the opinions of the whole school, would it?
That is why random sampling (choosing at random) is essential.
Common methods include using "random dice," a "random number table," or in recent times, a computer's "random number" function. The goal is to pick items so that no human intention influences the result.
【Common Mistake】
"Picking 10 of my close friends" is not random! That is called "intentional" or "biased" sampling, and it causes personal preferences to contaminate the survey results.
4. Predict the Total Using Calculation!
This is the part that often shows up on tests! We use the idea that "the ratio of the sample is approximately equal to the ratio of the population" to set up a proportion.
【Formula-like Approach】
\( \text{Proportion in the sample} = \text{Proportion in the population} \)
It’s easy to understand when written as a ratio:
(Specific count in sample) : (Sample size) = (Estimated count in population) : (Population size)
【Let's try an example!】
There are many white balls in a bag. We add 50 black balls and mix them well. We draw 40 balls and find that 4 of them are black. Roughly how many white balls were in the bag originally?
(Step 1) Set up the proportion
Focusing on the number of black balls, use \( (\text{Black in sample}) : (\text{Sample total}) = (\text{Black in total}) : (\text{Total population}) \).
\( 4 : 40 = 50 : x \)
(Step 2) Calculate
\( 4x = 40 \times 50 \)
\( 4x = 2000 \)
\( x = 500 \)
(Step 3) How to state your answer
The total population is 500. Since we know there are 50 black balls, the white balls are \( 500 - 50 = 450 \).
Since the result of a sample survey is an "estimate," always include the word "roughly" or "approximately" in your answer.
Answer: Roughly 450 balls
【It might feel difficult at first, but you'll be fine!】
If you always keep the form "(Part) : (Total of part) = (Part of total) : (Total of total)" in mind, you will be able to solve any problem.
5. Summary of Sample Surveys
Finally, let's review the key points of this chapter.
・When it's hard (or impossible) to check everything, use a "Sample Survey"!
・When selecting, be "random" to avoid bias!
・The larger the sample size (number of people, etc.), the higher the precision (accuracy) of the estimate!
・Use the proportion \( a : b = c : d \) to solve the calculation!
A sample survey is like a piece of "mathematical magic" that allows us to count things that are otherwise uncountable.
Next time you hear the term "opinion poll" on the news, try to remember: "Oh, that's a sample survey using random selection!" Great work today!