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The DailySTEP 4
Medium×2♥♥♥40
If 2x − 7 = 11, what is the value of 3x?
A9
B18
C27
Best: 3
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Expression of IdeasTransitions
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
Why: the second sentence adds another accomplishment. Nothing is being contrasted.
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98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.
Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.
Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.
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Inference and margin of error · Lesson 1
From a random sample to the whole population
A population is the whole group you want to know about, such as all 2,500 students at a school. A sample is the part of it you actually ask or measure. When the sample is chosen at random, every member of the population has a fair chance to be picked, so the sample tends to look like the population. That is what lets you use it to estimate.
Scaling a proportion. Suppose a random sample of 60 of those students finds that 18 of them play an instrument. The sample proportion is 6018=0.3, or 30%. The best estimate for the whole school is the same proportion of the population: 0.3×2,500=750 students.
Scaling a mean. If a random sample of a town's 4,000 households has a mean of 1.5 cars per household, the best estimate of the total number of cars in those households is 1.5×4,000=6,000. A mean is already per member, so you multiply it by the number of members in the population, not by the sample size.
Use careful language. 750 is the best estimate, not a promise: a different random sample would most likely give a slightly different number. Lesson 2 shows how a margin of error describes that uncertainty.
Read the question's last line twice. It may ask about the group in the sample, the other group, a difference between two groups, or a count instead of a percent.
Worked example Easy
A random sample of 150 of the 1,800 students at a high school were asked how they get to school. Of the students in the sample, 42 said they walk.
Based on the sample, which of the following is the best estimate of the number of students at the high school who walk to school?
A42
B504Answer
C1,296
D1,758
How to solve it
The sample proportion who walk is 15042=0.28, or 28%.
The sample was chosen at random, so use the same proportion for the whole school: 0.28×1,800=504.
Check the size: 504 is 28% of 1,800, a bit more than a quarter of the school, which matches 42 out of 150.
Why each choice is right or wrong
A. Incorrect. 42 is the number of walkers in the sample of 150 only. The question asks about all 1,800 students, so scale up: 15042×1,800=504.
B. Correct. The sample proportion is 15042=0.28, and 0.28×1,800=504 students.
C. Incorrect. This is the estimate for students who do not walk: 150108=0.72 and 0.72×1,800=1,296. The question asks about the students who walk.
D. Incorrect. This subtracts the sample count from the school's size: 1,800−42=1,758. The 42 walkers are part of a sample of 150, so they must be scaled up, not subtracted.
Worked example Medium
A random sample of 250 of the 16,000 registered voters in a county were asked which of two candidates, Ruiz or Bell, they plan to vote for. In the sample, 95 chose Ruiz, 110 chose Bell and 45 were undecided.
Based on the sample, which of the following is the best estimate of how many more of the county's registered voters plan to vote for Bell than for Ruiz?
A15
B960Answer
C6,080
D7,040
How to solve it
In the sample, Bell has 110−95=15 more supporters than Ruiz.
As a share of the sample, that difference is 25015=0.06.
Scale to the county: 0.06×16,000=960.
The long way gives the same answer: Bell 250110×16,000=7,040, Ruiz 25095×16,000=6,080, and 7,040−6,080=960.
Why each choice is right or wrong
A. Incorrect. 15 is the difference within the sample of 250. It has to be scaled to all 16,000 voters: 25015×16,000=960.
B. Correct. The difference in the sample is 110−95=15 voters, which is 25015=0.06 of the sample, and 0.06×16,000=960.
C. Incorrect. 6,080 is the estimate for Ruiz alone, 25095×16,000. The question asks how many more plan to vote for Bell.
D. Incorrect. 7,040 is the estimate for Bell alone, 250110×16,000. Subtract the estimate for Ruiz, 6,080, to find how many more plan to vote for Bell.
Trap Stopping at the sample
A choice often matches a number from the sample itself, such as the count of people who answered yes. The question asks about the population, so the sample count has to be scaled up to the population's size.
Trap Scaling the wrong group
Surveys have more than one category. If the question asks how many do not own a bike, or how many more chose one option, a choice built from the wrong category may be waiting. Underline the group the question names before you compute.
Trap Scaling a mean like a count
A mean is already per member, so multiply it by the population size. Dividing it by the sample size first, or multiplying by the sample size as well, gives a number that describes nothing in the problem.
Desmos When Desmos isn’t faster
Desmos doesn't speed up these questions. There is nothing to graph: the work is one proportion and one multiplication.
You can still use it as a calculator to avoid slips, for example by typing (42/150)*1800 on a line to see 504. A handheld calculator or quick mental math does the same job just as fast.
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