Worked Examples: SAT Strategy
Example 1: Easy (Margins of Error and Plausible Values)
Question: A researcher estimates that $49%$ of a population has a certain characteristic, with a margin of error of $4%$. Which of the following is the most appropriate conclusion?
A) Exactly $49%$ of the population has the characteristic.
B) It is plausible that the proportion is between $0.45$ and $0.53$.
SAT Strategy: A margin of error creates a "plausible interval" around the sample statistic. It is never a guarantee of an exact number.
- Start with the estimated mean/proportion: $49%$.
- Subtract the margin of error to find the lower bound: $49% - 4% = 45%$.
- Add the margin of error to find the upper bound: $49% + 4% = 53%$.
- The true population proportion is likely between $45%$ and $53%$.
- Answer: B.
Example 2: Medium (Calculating Residuals)
Question: A scatterplot shows the relationship between two variables, $x$ and $y$. A line of best fit is also shown. For the data point at $x = 5$, the actual $y$-value is $14$. The line of best fit predicts the $y$-value to be $10$. What is the residual for this data point?
SAT Strategy: The SAT heavily tests the concept of a "residual." The formula you must memorize is:
Residual = Actual Value - Predicted Value.
- Identify the Actual Value (the coordinate of the dot on the graph): $14$.
- Identify the Predicted Value (the $y$-value on the line of best fit): $10$.
- Subtract Predicted from Actual: $14 - 10 = 4$.
- Because the actual point is above the line of best fit, the residual is positive.
- Answer: $4$.
Example 3: Hard (Generalization and Causation Traps)
Question: A researcher wants to determine if listening to classical music improves memory. She asks 50 students from the local university's music department to volunteer. She then randomly assigns 25 to listen to classical music and 25 to sit in silence while memorizing a list. The music group performs significantly better. What conclusion can be drawn?
SAT Strategy: Break the study design into two parts: Selection and Assignment.
- Selection: Were the participants randomly selected from the general population? No, they were volunteers from a specific university's music department. Therefore, we cannot generalize this finding to all people, nor even all students. It only applies to students like those in the music department.
- Assignment: Were they randomly assigned to the music and silence groups? Yes. Random assignment separates the groups evenly, removing confounding variables. Therefore, we can claim that the music caused the improved memory for this specific group.
- Answer: Listening to classical music causes an improvement in memory for students similar to those in the local university's music department.
Practice Questions
Level 1: Easy (Foundational SAT)
- A poll of 500 likely voters in a city found that $52%$ support a new tax measure. The poll has a margin of error of $\pm 3%$. Based on this poll, which of the following statements is true?
- A) Exactly $52%$ of all city residents support the tax measure.
- B) The true percentage of likely voters who support the measure is likely between $49%$ and $55%$.
- C) $3%$ of the voters surveyed did not understand the question.
- D) If the poll were conducted again, exactly $52%$ of voters would support it.
- A line of best fit on a scatterplot predicts a value of $y = 25$ when $x = 4$. The actual data point on the scatterplot at $x = 4$ is $(4, 30)$. What is the residual for this data point? (Grid-in)
- A biologist studies the relationship between the temperature of an incubator and the growth rate of a specific bacteria. The scatterplot of the data shows a line of best fit with a negative slope. What does this indicate?
- A) As temperature increases, the growth rate increases.
- B) As temperature increases, the growth rate decreases.
- C) Temperature and growth rate are not related.
- D) The bacteria died before the experiment finished.
- The equation of a line of best fit is $y = 3.2x + 15$, where $x$ represents years of experience and $y$ represents hourly wage in dollars. What is the best interpretation of the number 3.2 in this context?
- A) The starting hourly wage for a worker with zero years of experience.
- B) The estimated increase in hourly wage for each additional year of experience.
- C) The exact hourly wage of a worker with 1 year of experience.
- D) The number of years required to earn 15 dollars an hour.
- A researcher randomly selects 100 dogs from a specific animal shelter to participate in a dietary study. Can the results of this study be generalized to all dogs in the world?
- A) Yes, because the dogs were randomly selected.
- B) No, because the sample size is too small.
- C) No, because the sample was only selected from one specific shelter.
- D) Yes, because dogs generally share the same dietary needs.
Level 2: Medium (Standard SAT)
- A botanist estimates the mean height of a species of plant in a forest to be 45 cm, with a margin of error of 4 cm. Which of the following is a plausible value for the true mean height of this plant species in the forest?
- A) 39 cm
- B) 40 cm
- C) 43 cm
- D) 50 cm
- The line of best fit for a set of data is $y = -0.5x + 20$. A data point in the set has an $x$-value of 6 and a residual of $-2$. What is the actual $y$-value of this data point? (Grid-in)
- Two surveys are conducted to estimate the proportion of students who drive to school. Survey A samples 50 students and has a margin of error of $8%$. Survey B samples 500 students. Assuming both surveys used the same random sampling methods, what is most likely true about Survey B?
- A) Survey B will have a margin of error greater than $8%$.
- B) Survey B will have a margin of error less than $8%$.
- C) Survey B will have a margin of error of exactly $8%$.
- D) Survey B will be completely free of any error.
- A gym owner wants to know if members are satisfied with the new equipment. She asks the first 40 members who walk into the gym on a Monday morning. She finds that $90%$ are satisfied. What is the most significant flaw in her study design?
- A) She didn't establish a control group.
- B) The sample was not randomly selected, so it may not represent all gym members.
- C) The survey question was likely too confusing.
- D) She didn't randomly assign members to different pieces of equipment.
- A scatterplot maps the number of hours studied ($x$) against test scores ($y$). The line of best fit always passes exactly through which specific coordinate point?
- A) $(0, 0)$
- B) The $y$-intercept
- C) The mean of the $x$-values and the mean of the $y$-values: $(\bar{x}, \bar{y})$
- D) The median of the $x$-values and the median of the $y$-values
Level 3: Hard (Advanced SAT)
- A medical researcher is testing a new allergy medication. She recruits 200 volunteers who suffer from severe allergies. She randomly assigns 100 to take the new medication and 100 to take a placebo. The group taking the new medication reports a significant reduction in symptoms. Which of the following is the most appropriate conclusion?
- A) The new medication causes a reduction in allergy symptoms for all people.
- B) The new medication causes a reduction in allergy symptoms for people with severe allergies similar to the volunteers.
- C) There is a positive association between the medication and symptom reduction, but causation cannot be proven because they were volunteers.
- D) The medication is guaranteed to work for the volunteers in the future.
- In a study of house prices, the line of best fit is determined to be $y = 150x + 50,000$, where $x$ is the area of the house in square feet and $y$ is the predicted price in dollars. A house with an area of 2,000 square feet is sold for $\$320,000$. What is the absolute value of the residual for this house? (Grid-in)
- Which of the following research designs would best allow a scientist to determine whether a new type of fertilizer causes plants to grow taller?
- A) Observe two different farms, one that uses the fertilizer and one that does not, and measure the plants.
- B) Randomly select 100 plants from a greenhouse that already uses the fertilizer and measure their heights.
- C) Take 100 identical plants, randomly assign 50 to receive the fertilizer and 50 to receive no fertilizer, and measure their heights.
- D) Ask farmers who use the fertilizer to report how tall their plants grow compared to previous years.
- A wildlife biologist estimates the average weight of an adult male black bear in a specific national park to be 250 lbs, with a margin of error of 15 lbs at a $95%$ confidence level. Which of the following statements is mathematically true regarding this estimate?
- A) $95%$ of all adult male black bears in the park weigh between 235 lbs and 265 lbs.
- B) It is highly plausible that the true average weight of the population of adult male black bears in the park is between 235 lbs and 265 lbs.
- C) If the biologist caught one more bear, there is a $95%$ chance it would weigh exactly 250 lbs.
- D) The biologist is absolutely certain the average weight is at least 235 lbs.
- A scatterplot shows a set of data points $(x,y)$ and a line of best fit. If every single $y$-value in the dataset is increased by 10, but the $x$-values remain identical, what will happen to the residuals of the data points?
- A) The residuals will all increase by 10.
- B) The residuals will all decrease by 10.
- C) The residuals will remain exactly the same.
- D) The residuals will be multiplied by 10.
Level 4: Very Hard (Slightly beyond SAT)
- A data scientist creates a line of best fit mapping the relationship between daily ice cream sales ($x$) and daily umbrella sales ($y$) in a city. The line of best fit has a strong positive slope. The data scientist concludes that buying ice cream causes people to buy umbrellas. What is the fatal flaw in this conclusion?
- A) The sample size of the days measured wasn't provided.
- B) It confuses correlation with causation; a third confounding variable, like hot weather, likely causes both.
- C) Ice cream and umbrellas are unrelated products, so the line of best fit must be calculated incorrectly.
- D) The margin of error for umbrella sales is likely too high.
- The line of best fit for predicting a student's final exam score ($y$) based on the number of hours they studied ($x$) is $y = 4.5x + 40$. A student who studied for 8 hours received a final exam score of 72. What is the residual for this student? (Grid-in)
- A university surveys a random sample of 200 freshmen about their sleep habits. The university reports that freshmen sleep an average of 6.2 hours per night, with a margin of error of 0.5 hours. To decrease the margin of error to 0.25 hours (cut it in half) while maintaining the same confidence level, what must the university do?
- A) Divide the sample size in half (survey 100 students).
- B) Double the sample size (survey 400 students).
- C) Quadruple the sample size (survey 800 students).
- D) Survey only students who sleep more than 6 hours.
- An exponential regression curve $y = a(b)^x$ is used to model population growth over time. For the data point at $x = 2$, the predicted $y$-value is 100. If the residual for this point is 15, what is the actual population measured at $x=2$? (Grid-in)
- A researcher randomly assigns 60 high schools across the state into two groups to test a new math curriculum. Group A uses the new curriculum, and Group B uses the old curriculum. At the end of the year, Group A scores significantly higher on state tests. Which of the following is the most scientifically sound conclusion?
- A) The new curriculum will cause higher math scores for any high school student in the world.
- B) The new curriculum caused the higher math scores for the high schools in the state.
- C) There is a correlation, but not causation, because the students within the high schools were not individually randomized.
- D) The old curriculum causes students' math abilities to decrease over time.
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