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If 2x − 7 = 11, what is the value of 3x?
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Jazz pianist Mary Lou Williams composed hundreds of works. ______ she wrote arrangements for bands led by Duke Ellington and Benny Goodman.
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Inference and margin of error · Lesson 2
What a margin of error means
A random sample gives an estimate, but a different random sample would give a slightly different one. The margin of error describes how far the estimate could plausibly be from the true population value.
Subtract it and add it. If a survey of a random sample of a city's adults estimates that 52% of them favor a later school start time, with a margin of error of 3%, the interval runs from 52−3=49 to 52+3=55, in percent. For a percent, the margin is in percentage points: 3% here means 3 points, not 3% of 52.
The SAT's wording for what this means: it is plausible that between 49% and 55% of all the city's adults favor a later start. Values inside the interval are plausible. Values outside it are less plausible, and the further outside, the less plausible.
The same reading works for a mean. If a random sample of a school's students spends a mean of 6.2 hours a week on homework, with a margin of error of 0.4 hours, it is plausible that the mean for all the school's students is between 5.8 and 6.6 hours. That says nothing about how long any one student spends.
A margin of error is reported at a confidence level, often 95%, so even the interval isn't a guarantee. What the SAT asks you to know is what the interval is about: the population's percent or mean, as a range of plausible values. The margin covers only the chance variation of random sampling (which members happened to be picked), not bias from a nonrandom sample or a misleading question.
What an estimate with a margin of error supports
Supported
Not supported
The true value is plausibly in the interval
The true value is exactly the estimate
A claim about the population's percent or mean
A claim about individual members
Plausible, reasonable to conclude
Certain, proven, guaranteed
The margin covers chance variation from random sampling
The margin covers bias from a nonrandom sample or a misleading question
Worked example Easy
A survey of a random sample of a city's registered voters found that 58% of those surveyed support building a new library. The margin of error for the survey is 4%.
Which of the following is the most appropriate conclusion about all registered voters in the city?
AExactly 58% of all the city's registered voters support building a new library.
BIt is plausible that 54% to 62% of the city's registered voters support it.Answer
CIt is certain that more than half of the city's registered voters support building it.
DThe margin of error shows that 4% of the voters surveyed gave inaccurate answers.
How to solve it
Build the interval: 58−4=54 and 58+4=62, so 54% to 62%.
The interval gives plausible values for the population, all the city's registered voters.
Of these four choices, only B uses the whole interval, names all the city's registered voters and says plausible, so it is the only one that says no more than the survey shows.
Why each choice is right or wrong
A. Incorrect. 58% is the estimate. The margin of error says the true percent could plausibly be anywhere from 54% to 62%, so exactly 58% claims more than the survey shows.
B. Correct. 58%±4% gives the interval 54% to 62%, and plausible is the right strength for a claim about the population.
C. Incorrect. The whole interval is above 50%, so a majority is a reasonable conclusion, but no sample makes it certain. The word certain claims more than a survey can show.
D. Incorrect. The margin of error isn't a count of wrong answers. It describes how far an estimate from a random sample could plausibly be from the population value just because of which people happened to be chosen.
Worked example Medium
A random sample of 200 adults who live in a county and commute to work were asked how long their commute takes. The mean commute time for the sample was 27 minutes, with a margin of error of 3 minutes.
Which of the following is the most appropriate conclusion?
AMost of the county's adults who commute to work have a commute that takes between 24 and 30 minutes.
BThe mean commute time of all the county's adults who commute to work is exactly 27 minutes.
CThe mean commute time of all the county's adults who commute to work is plausibly between 24 and 30 minutes.Answer
DThe mean commute time of all the state's adults who commute to work is plausibly between 24 and 30 minutes.
How to solve it
Build the interval: 27−3=24 and 27+3=30 minutes.
The interval is about the mean commute of the population the sample came from: the county's adults who commute to work.
Check each choice for the right quantity (the mean, not individuals), the right group (the county's commuters) and the right strength (plausibly, not exactly).
Why each choice is right or wrong
A. Incorrect. The interval is about the mean commute, not individual commutes. Individual commutes can spread far wider, from a few minutes to over an hour, while the mean stays near 27 minutes.
B. Incorrect. 27 minutes is the sample mean. The margin of error of 3 minutes says the mean for the county's commuters could plausibly be anywhere from 24 to 30 minutes.
C. Correct. 27±3 gives 24 to 30 minutes. The claim is about the mean, for exactly the population the sample came from, the county's adults who commute to work, and it says plausibly.
D. Incorrect. The sample was chosen from commuters in the county, so the conclusion applies only to the county. Commuters elsewhere in the state weren't sampled.
Trap Treating the estimate as the exact answer
Exactly 58% or the true mean is 27 minutes states the estimate as a fact. The true value is one number you don't know. The margin of error gives a range of plausible values for it, and the estimate is only the middle of that range.
Trap Reading a mean's interval as a range for individuals
An interval for a mean describes the population's average, not where each member falls. Most students spend between 5.8 and 6.6 hours on homework is a different claim, and the survey doesn't support it.
Trap Certainty words
Certain, proves and guarantees claim more than a random sample can show. A correct conclusion never overclaims. It may hedge with words such as plausible, reasonable to conclude or likely, or it may say what the data don't show. Careful words alone don't make a choice right, though: many wrong choices use them too. When a choice states a value for the population, the group, the quantity (a mean or percent, not individual values) and the numbers also have to match what was sampled, unless the choice is pointing out that the sample is biased.
Desmos When Desmos isn’t faster
Desmos isn't faster here. These are reading questions: the arithmetic is one subtraction and one addition, and the real work is matching the conclusion to what the interval describes.
If the interval has to become a count, such as 54% to 62% of 15,000 voters, you can type 0.5415000 and 0.6215000 on two lines, but mental math or any calculator is just as quick.
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