If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Course: Digital SAT Math>Unit 7

Lesson 9: Data inferences: medium

Data inferences | Lesson

A guide to data inferences on the digital SAT

What are data inferences questions?

When we want to answer questions like "how many voters feel positively about a new law" or "what percentage of Americans exercise regularly", it's often impractical to ask everyone—it would take a lot of time and effort to ask every voter, let alone every American!
Instead, when we have questions about a large population, we often answer those questions by surveying a representative sample: a smaller set of people whose answers can give us a good idea of how the population would answer the same questions.
In this lesson, we'll learn to:
1. Make generalizations about a population based on sampling data
2. Use margin of error to describe the uncertainty of sampling
You can learn anything. Let's do this!

How do I make generalizations about a population using sampling data?

Estimating using sample proportions

A random sample drawn from a population is representative of the population. With a representative sample, we can multiply the
by the population to get an estimate.
$\text{estimate}=\text{sample proportion}\cdot \text{population}$

Let's look at some examples!

A representative sample of households in City A reveals that $14.8\mathrm{%}$ of the households in the sample have exactly two children under the age of $18$. If City A has a total of $59,317$ households, approximately how many of them have exactly two children under the age of $18$ ?

$50$ seniors at a particular high school are randomly selected for a survey. $21$ of them report riding their bikes to school at least once a week. If there are $300$ seniors at this high school, what is a reasonable estimate of the total number of seniors who ride their bikes to school at least once a week?

Try it!

try: use sample data to make a prediction
An inspector examined a random sample of $500$ bars of dark chocolate at the Nomnom Chocolate Factory and found $2$ of the bars to be defective. The factory produces $60,000$ bars of dark chocolate a week.
Write the sample proportion of the number of defective bars to the total number of bars as a fraction:
At this rate, how many defective dark chocolate bars is the Nomnom Chocolate Factory expected to produce in a week?

What is margin of error?

While we can make reasonable estimates using sample proportions, we can never be $100\mathrm{%}$ certain that the population proportion matches the sample proportion exactly. Margins of error let us address the uncertainty inherent to sampling.
The margin of error is most commonly given as a percentage. When given a percent estimate and a margin of error, we can establish a range around the estimate by adding and subtracting the margin of error.
$\text{range}=\text{estimate}±\text{margin of error}$
For example, if a poll estimates that a political candidate will win $51\mathrm{%}$ of the popular vote with a margin of error of $2\mathrm{%}$, what it actually means is that the poll is reasonably sure that the candidate will actually win $51\mathrm{%}±2\mathrm{%}$ of the popular vote, or anywhere between $49\mathrm{%}$ and $53\mathrm{%}$.
Note: in the above example, there's still no $100\mathrm{%}$ certainty that the candidate will win between $49\mathrm{%}$ to $53\mathrm{%}$ of the popular vote! However, the poll can be more confident in the $51\mathrm{%}±2\mathrm{%}$ estimate than in the exact $51\mathrm{%}$ estimate.
The larger a sample size is, the smaller the margin of error will be. Think about it this way: if we want to make an estimate about a population of a million people, we'll get a more accurate result if we ask a random sample of $5000$ people than if we ask only a random sample of $50$.

Try it!

try: use margin of error to draw a conclusion
A researcher surveyed a random sample of students from a large university about how often they talk to their parents. Using the sample data, the researcher estimated that $12\mathrm{%}$ of the students in the population talk to their parents at least once per day. The margin of error for this estimation is $3\mathrm{%}$.
The researcher is reasonably confident that between
of the students at the university talk to their parents at least once per day.
If the researcher doubled the size of the random sample, the margin of error would likely
.

Practice: use sample rate to make a prediction
In a recent survey of $600$ randomly selected registered voters in the town of Carrington, $482$ are in favor of increasing funding for the town's mental health services. Based on the survey results, approximately how many of Carrington's $19,310$ registered voters are in favor of increasing funding for the town's mental health services?

Practice: draw conclusion based on margin of error
Anya surveyed a random sample of members of a large gym about how often they visit the gym. Using the sample data, she estimated that $74\mathrm{%}$ of gym members go to the gym at least once a week, with a margin of error of $3\mathrm{%}$. Which of the following is the most appropriate conclusion about all members of the gym, based on the given estimate and margin of error?

Things to remember

$\begin{array}{rl}& \text{estimate}=\text{sample proportion}\cdot \text{population}\\ \\ & \text{range}=\text{estimate}±\text{margin of error}\end{array}$
The larger a sample size is, the smaller the margin of error will be.

Want to join the conversation?

• I feel like this all feels too easy when you're practicing but when you give the actual test ,the score is always underwhelming
• but this is because you didn't studied or because the exercises are too lower level compared to the questions of the test?
• I finished this lesson in 14±2 minutes
• It seems SAT Math is easy than our A levels
• Yes, honestly, I'm studying for my Nigerian JAMB exams and SAT math is by far easier and less broad than JAMB or even my school A Level Math.
• Will we be always given the margin of error on the SAT or will we ever have to calculate it?
• I always see your comments on the videos/articles. Have you finished the DSAT math and/or reading course? If so, has it helped you improve your score?
• how to calculate the margin of error if asked in the test?
• What is this sign? ±
(1 vote)
• It means Plus or Minus. 2±1 could equal both 2+1=3 and 2-1=1