SAT probability and statistical inference: review and practice

• Find a probability from counts, a frequency table or a two-way table.
• Find a conditional probability, such as the chance that a randomly chosen senior plays a sport.
• Use a sample result and its margin of error to give a plausible range for the whole population.
• Decide which population a random sample's results apply to.
• Judge whether a study supports a cause-and-effect claim or only an association.

• Dividing by the grand total on a "given that" question. If 30 of 80 seniors play a sport, the probability that a senior plays a sport is 30/80, not 30 divided by everyone in the table.
• Thinking a bigger sample fixes a biased one. Surveying 5,000 people at a gym still tells you about gym members, not the whole city.
• Confusing random sampling with random assignment. Random sampling lets you generalize to the population; random assignment to treatments lets you claim cause. A study can have one without the other.
• Generalizing past the population sampled. A random sample of one school's juniors supports a conclusion about that school's juniors, not all high school students.
• Applying the margin of error to the wrong number. An estimate of 58% with a margin of 4 percentage points gives 54% to 62%. To estimate a count, multiply both ends by the population size.
• Rejecting a study for an irrelevant reason. Unequal group sizes alone don't make a randomized experiment invalid.

1. Question 1 · Easy

   A total of 325 visitors are viewing exhibits at a museum. Each visitor is in either the ceramics gallery, the textile gallery, or the sculpture gallery. If one of these visitors is selected at random, the probability that the visitor is in the ceramics gallery is 0.72 and the probability that the visitor is in the textile gallery is 0.20. How many visitors are in the sculpture gallery?

   • A8
   • B26
   • C65
   • D169
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   B. The remaining probability is 1 − 0.72 − 0.20 = 0.08, so the number of people in the sculpture gallery is 325(0.08) = 26. Choice A treats the remaining percentage number, 100(1 − 0.72 − 0.20) = 8, as a count; choice C counts the second named group, 325(0.20) = 65; choice D uses the difference between the known groups, 325(0.72 − 0.20) = 169.

2. Question 2 · Easy

   Finish = Matte, Floral = 34, Geometric = 25, Landscape = 32. Finish = Glossy, Floral = 26, Geometric = 41, Landscape = 22. The table shows the postcards available in a museum shop, classified by finish and design. What fraction of the postcards with a floral design have a matte finish?

   • A34/180
   • B34/91
   • C91/180
   • D34/60
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   D. There are 34 + 26 = 60 postcards with a floral design, and 34 have a matte finish, so the fraction is 34/60. Choice A divides by all 180 postcards; choice B divides by the 91 matte postcards; choice C gives the fraction of all postcards that are matte.

3. Question 3 · Easy

   Item type = Hardcover book, Frequency = 12. Item type = Paperback book, Frequency = 34. Item type = Magazine, Frequency = 50. The frequency table shows the types of items returned to a library on one day. If one of these items is selected at random, what is the probability that the item is a hardcover book?

   • A12/96
   • B12/84
   • C12/50
   • D12/34
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   A. The table contains 12 + 34 + 50 = 96 returned items, including 12 hardcover books, so the probability is 12/96 = 1/8. Choice B uses only the other two categories in the denominator, 12/(34 + 50) = 12/84; choice C uses the magazine count as the denominator, 12/50; choice D uses the paperback count as the denominator, 12/34.

4. Question 4 · Medium

   A city has 24,000 adult residents. A community planning office surveyed 1,600 of these residents chosen at random about a grant program for neighborhood gardens. The office estimated that 64% of all adult residents support the program, with a margin of error of 1.5 percentage points. Based on these results, which of the following is a plausible value for the total number of adult residents who support the grant program?

   • A576
   • B1,024
   • C14,900
   • D15,480
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   D. The percentage interval is 64 − 1.5 = 62.5 to 64 + 1.5 = 65.5, giving counts from 0.625 × 24,000 = 15,000 to 0.655 × 24,000 = 15,720, which include 15,480. Choice A uses 36% of the sample; choice B uses 64% of the sample; choice C falls below 15,000.

5. Question 5 · Medium

   Station = Central, Number of selected commuters = 34. Station = Riverside, Number of selected commuters = 46. Station = Uptown, Number of selected commuters = 61. A transportation researcher studies the relationship between commute distance and weekly travel cost for adult train commuters in a state. The researcher randomly selects commuters from passenger lists at three stations in one metropolitan area. All 141 selected commuters report their travel distances and costs each week for six weeks. The table shows how many participants were selected at each station. The researcher finds an association between the two variables and considers using the findings to describe commuters throughout the state. Which of the following aspects of the study will prevent the results from being generalized to all adult train commuters in the state?

   • AThe sampling lists covered passengers at only three stations in one metropolitan area.
   • BThe six-week observation period was too short to study this association.
   • CThe numbers of participants selected at the three stations were unequal.
   • DA sample of 141 commuters was below the minimum needed to study this association.
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   A. The passenger lists omit commuters elsewhere in the state, so random selection from those lists does not justify a statewide generalization. Choice C treats unequal subgroup counts as a coverage problem, while choices D and B assert sample-size and duration requirements that were not provided and would not repair the geographic gap.

6. Question 6 · Medium

   Age (years) = Under 18, Percentage of all residents = 31%. Age (years) = 18–44, Percentage of all residents = 23%. Age (years) = 45–69, Percentage of all residents = 27%. Age (years) = 70 or older, Percentage of all residents = 19%. The table gives the percentage of residents in each age group for a town. One of the town’s residents is selected at random. Which of the following is closest to the probability that the selected resident is at least 70 years old, given that the resident is at least 18 years old?

   • A0.19
   • B0.28
   • C0.50
   • D0.61
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   B. Residents aged at least 18 account for 23 + 27 + 19 = 69 percent, so the requested probability is 19/69 ≈ 0.28. Choice A uses the oldest group’s percentage of all residents, choice C adds the two other adult percentages without renormalizing, and choice D divides by the under-18 percentage.

Type a probability as a fraction, like 30/80, to get the decimal and compare it with the choices. For a margin of error, type each endpoint on its own line, such as 0.54 × 12000 and 0.62 × 12000, to turn a percent range into a range of counts. Desmos can't tell you which conclusion a study supports; that comes from reading how the sample was chosen and whether treatments were randomly assigned.

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