On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
My approach to this kind of question starts with a pot of rice. When my mother cooked, she did not eat the whole pot to know if it was ready. She took one spoon, tasted it, and decided about the whole pot. She also knew the spoon might be a little different from the rest, so she stirred well first. That is exactly sampling. A small, fair taste tells you about the whole thing, with a small possible error. On the SAT, the spoon is the sample and the pot is the population. The margin of error tells you how far the pot might be from your spoon. I ask two things: was the spoon taken fairly, and how big is the wiggle room?
This skill is mostly reading carefully and adding or subtracting one number. It looks like statistics, but it is just a spoon and a pot. You will get it.
What is inside this part
Using the result from a random sample to estimate a percent, mean or count for the whole population.
Sample proportion times population size · Applying a sample percent to the population
Finding the range of plausible population values by going down and up from the sample result by the margin of error.
Result minus margin to result plus margin · Which value is or is not plausible
Understanding how the size of the sample and the confidence level change the margin of error.
Bigger sample gives smaller margin · Square root effect
Choosing the statement that the sample really supports, and no more.
Right population · Mean versus individuals · Plausible versus exactly
How to know a question is from this part
A survey or sample result with a margin of error, and a question about the larger group.
- The words 'random sample' and the size of the whole population
- 'Margin of error' or 'confidence level'
- 'Best estimate' or 'about how many'
- 'Which conclusion is best supported' or 'plausible value'
- A pair of samples of different sizes to compare
The lesson
1. Population and sample
The population is the whole group we want to know about: all students in a school, all trees on a farm, all voters in a city. The sample is a small part of that group that we actually check.
We use a sample because checking everyone takes too much time and money. The statistic is the number we find from the sample, like a percent or a mean. The goal is to use the statistic to make an inference (a careful guess) about the population.
A school has 2,000 students. We ask 200 of them about their favourite sport. Which is the population and which is the sample?
Show the steps
- Population: all 2,000 students.
- Sample: the 200 students we asked.
Population 2,000, sample 200
2. When can we trust a sample?
A sample is good for inference when it is random. A random sample gives every member of the population an equal chance to be chosen. Then the sample is likely to look like the population.
The SAT almost always says 'random sample'. When it does, you may make a conclusion about the whole population. If the sample is NOT random, or comes from only one part of the population, the conclusion cannot be stretched to everyone.
Also, we can only make a conclusion about the population the sample came from. A sample of students in one town tells us about that town, not about the whole country.
3. Using a sample proportion to estimate a count
A proportion is a part divided by the whole, like 50 out of 200. First find the sample proportion. Then multiply it by the size of the population.
This gives an estimate, not an exact count. The SAT says 'best estimate' or 'about'.
In a random sample of 400 residents of a town of 25,000, 168 support a plan. Estimate how many of the 25,000 support it.
Show the steps
- Sample proportion = 168/400 = 0.42.
- Multiply by the population: 0.42 × 25,000.
- = 10,500.
10,500
4. What margin of error means
A sample is not perfect, so its result is a little off. The margin of error tells us how far off it could be. It is the plus-or-minus number.
If the sample says 62% and the margin of error is 4 percentage points, then the true value for the population is plausibly between 62 - 4 and 62 + 4. That range, 58% to 66%, is the interval.
Notice the word plausible. The true value is probably in the interval, but we can never be 100% sure. Values outside the interval are not plausible.
A random sample gives a mean of 14.2 minutes with a margin of error of 1.1. Which value is plausible for the population mean: 12.4, 15.0 or 16.2?
Show the steps
- Low end: 14.2 - 1.1 = 13.1.
- High end: 14.2 + 1.1 = 15.3.
- Only 15.0 is between 13.1 and 15.3.
15.0
5. Confidence level and sample size
The confidence level (like 95%) tells how sure the method is. A higher confidence level makes the margin of error bigger, because to be surer you need more wiggle room.
A larger sample makes the margin of error smaller, because more information gives a closer estimate. The size effect is by the square root: to cut the margin in half you need 4 times the sample. To cut it to a third you need 9 times.
The SAT usually asks which sample has the smaller margin, or about how big a sample is needed. It does not ask you to compute a margin of error from scratch.
Sample A: 100 people, margin 10 points. Sample B: 400 people, same confidence. Margin of B?
Show the steps
- B is 4 times bigger.
- Square root of 4 is 2.
- Margin = 10/2 = 5 points.
5 points
6. Reading conclusions carefully
The margin of error is about the population value (the mean or the percent), not about every individual. A statement such as 'every plant is between 59 and 65 cm' is wrong, because a single plant can be very different from the mean.
If the whole interval is above 50%, you can say a majority likely approves. If the interval includes 50%, you cannot be sure about a majority, and 50% is a plausible value.
Do not turn 'plausible' into 'exactly' or 'certainly'. Do not apply the result to a different population than the one sampled.
A random sample of voters: 54% approve, margin of error 5 points. Is a majority sure to approve?
Show the steps
- Interval: 49% to 59%.
- 50% is inside the interval.
- So a majority is possible but not certain.
No, 50% is plausible
7. Common traps
Trap 1: Forgetting to subtract as well as add the margin of error.
Trap 2: Using the sample count when the question asks about the population.
Trap 3: Saying the margin shrinks in a straight line with sample size. It shrinks with the square root.
Trap 4: Stretching the conclusion to a bigger group than the one sampled, or claiming 'exactly' or 'every'.
See it
- 1Check: is the sample random? From which population?
- 2Find the sample result (percent or mean).
- 3Interval = result - margin to result + margin.
- 4Is the value in the question inside the interval? If yes, it is plausible.
- 5Keep the conclusion about the same population and about the mean or percent only.
| If I... | The margin of error... | Why |
|---|---|---|
| Take a bigger sample | gets smaller | More information |
| Ask for a higher confidence level | gets bigger | More wiggle room for more certainty |
| Make the sample 4 times bigger | becomes half | It shrinks with the square root |
| Make the sample 9 times bigger | becomes one third | The square root of 9 is 3 |
- It is plausible the population mean is between 59 and 65.
- About 10,500 residents support the plan.
- A majority probably approve, since the whole interval is above 50%.
- Every plant is between 59 and 65.
- The mean is exactly 62.
- A majority approve, even though the interval includes 50%.
- Results for one city apply to the whole country.
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