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Data inferences | Lesson

What are data inferences questions, and how frequently do they appear on the test?

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 these 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
On your official SAT, you'll likely see 1 question that tests your ability to make inferences based on sampling data.
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.
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Let's look at some examples!

A representative sample of households in City A reveals that 14, point, 8, percent of the households in the sample have exactly two children under the age of 18. If City A has a total of 59, comma, 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, comma, 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:
  • Your answer should be
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
At this rate, how many defective dark chocolate bars is the Nomnom Chocolate Factory expected to produce in a week?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

What is margin of error?

Note: questions about margin of error appear very rarely on the test. When they do appear, the margins of error are given—we don't need to calculate them ourselves.
While we can make reasonable estimates using sample proportions, we can never be 100, percent 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.
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For example, if a poll estimates that a political candidate will win 51, percent of the popular vote with a margin of error of 2, percent, what it actually means is that the poll is reasonably sure that the candidate will actually win 51, percent, plus minus, 2, percent of the popular vote, or anywhere between 49, percent and 53, percent.
Note: in the above example, there's still no 100, percent certainty that the candidate will win between 49, percent to 53, percent of the popular vote! However, the poll can be more confident in the 51, percent, plus minus, 2, percent estimate than in the exact 51, percent 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, percent of the students in the population talk to their parents at least once per day. The margin of error for this estimation is 3, percent.
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

Your turn!

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, comma, 310 registered voters are in favor of increasing funding for the town's mental health services?
Choose 1 answer:

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, percent of gym members go to the gym at least once a week, with a margin of error of 3, percent. Which of the following is the most appropriate conclusion about all members of the gym, based on the given estimate and margin of error?
Choose 1 answer:

Things to remember

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

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