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The DailySTEP 4
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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.
AHowever,Your pick
BIn addition,Correct
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Inference and margin of error · Lesson 5
Comparing estimates with margins of error
Some questions give two estimates, each with its own margin of error, and ask whether one population value is greater than the other. Don't compare the estimates alone. Turn each into its interval first.
No overlap: if every plausible value for A is greater than every plausible value for B, it is reasonable to conclude that A's true value is greater. For example, 45%±3% (42% to 48%) and 38%±2% (36% to 40%) don't overlap, so it is reasonable to conclude that the first population's percent is the greater one.
Overlap: if the intervals overlap, the SAT treats the data as not showing which true value is greater, because a value inside the overlap is plausible for both. This is a cautious way to read intervals, not a proof that there is no difference: overlap doesn't show the values are equal either.
Comparing with a fixed value works the same way. On the SAT, a claim such as more than half of the voters support the measure is reasonable only if the whole interval is above 50%. An estimate of 53%±4% gives 49% to 57%, which includes values below 50%, so a majority isn't a reasonable conclusion.
Both estimates must come from random samples of the populations being compared, or the comparison isn't supported at all.
Worked example Medium
Random samples of adults in two towns were asked whether they use public transit at least once a week. In Town A, the estimate was 46% with a margin of error of 3%. In Town B, the estimate was 55% with a margin of error of 4%.
Which of the following is the most appropriate conclusion?
AIt is reasonable to conclude that a greater percent of Town B's adults use public transit weekly.Answer
BNo conclusion about the two towns is possible, because the two surveys have different margins of error.
CThe percent of Town B's adults who use public transit weekly is exactly 9 points higher than Town A's.
DIt is reasonable to conclude that the percents for the two towns are equal.
How to solve it
Town A: 46−3=43 to 46+3=49, in percent.
Town B: 55−4=51 to 55+4=59, in percent.
The lowest plausible value for Town B, 51%, is above the highest for Town A, 49%. The intervals don't overlap, so it is reasonable to conclude Town B's percent is greater.
Why each choice is right or wrong
A. Correct. Town A's interval is 43% to 49% and Town B's is 51% to 59%. They don't overlap, and every plausible value for Town B is higher.
B. Incorrect. Different margins of error don't block a comparison. Build each interval, then check for overlap; here there is none.
C. Incorrect. 9 points is the difference between the two estimates, 55−46. Each true percent is known only to within its margin of error, so the difference isn't known exactly.
D. Incorrect. The intervals don't overlap, so no single value is plausible for both towns. Equal percents aren't supported.
Worked example Hard
A survey of a random sample of a city's registered voters estimated that 51% of them support a proposed tax on sugary drinks, with a margin of error of 3%.
Which of the following is the most appropriate conclusion?
AA majority of the city's registered voters support the tax, because 51% is greater than 50%.
BFewer than half of the city's registered voters support the tax.
CThe survey doesn't show whether more than half of the city's registered voters support the tax.Answer
DExactly 51% of the city's registered voters support the proposed tax on sugary drinks.
How to solve it
Interval: 51−3=48 to 51+3=54, in percent.
The interval includes values below 50% and values above 50%.
So the survey supports neither a majority nor fewer than half.
Why each choice is right or wrong
A. Incorrect. This is the tempting choice. 51% is only the estimate. The interval, 48% to 54%, includes values below 50%, so a majority isn't a reasonable conclusion.
B. Incorrect. The interval also includes values above 50%, such as 53%, so the survey doesn't show that fewer than half support the tax either.
C. Correct. 51%±3% gives 48% to 54%. Plausible values fall on both sides of 50%, so the survey can't settle whether a majority supports the tax.
D. Incorrect. 51% is the estimate. With a margin of error of 3%, the true percent is plausibly anywhere from 48% to 54%, not exactly 51%.
Trap Comparing only the estimates
55% is more than 46% or 51% is more than half skips the margin of error. Turn each estimate into an interval before you compare.
Trap Reading overlap as equal
On the SAT, overlapping intervals mean the data don't show which value is greater. They don't show that the two values are the same, either.
Trap An interval that touches the line
An interval such as 50% to 56% includes 50% itself. On the SAT, more than half is a reasonable conclusion only when every plausible value is above 50%, so an interval that reaches down to exactly 50% doesn't support it.
Desmos When Desmos isn’t faster
Desmos isn't faster for comparing intervals. Each interval is one subtraction and one addition, and the comparison is reading two ranges.
A quick sketch on scratch paper works better: draw a number line, mark each interval, and look for overlap or for where 50% falls.
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