Worked Examples: SAT Strategy
Example 1: Easy (Interpreting the Model)
Question: A biologist studying the growth of a particular species of plant created a scatterplot showing the plant's height, $h$, in centimeters, $t$ days after it sprouted. The line of best fit for the data is given by the equation $h = 0.65t + 2.1$. What is the best interpretation of the number $2.1$ in this context?
SAT Strategy: Recognize the $y = mx + c$ structure. The $y$-intercept ($c$) always represents the starting value or initial condition when the $x$-variable is zero.
- Identify the variables: $t$ is days, $h$ is height.
- The number $2.1$ is the $h$-intercept, which occurs when $t = 0$.
- "When $t = 0$" means the exact day the plant sprouted.
- Answer: The predicted height of the plant, in centimeters, on the day it sprouted.
Example 2: Medium (Actual vs. Predicted / Residuals)
Question: A scatterplot relates two variables, $x$ and $y$. The line of best fit for the data is $y = 3.5x - 4$. One of the data points has an $x$-value of 6 and a $y$-value of 19. By how much does the actual $y$-value differ from the $y$-value predicted by the line of best fit?
SAT Strategy: The SAT loves asking for the difference between a specific data point (actual) and the line (predicted). Simply plug the $x$-value into the model to find the predicted value, then subtract it from the actual value.
- Find the predicted value: plug $x = 6$ into the line of best fit.
- $Predicted\ y = 3.5(6) - 4 = 21 - 4 = 17$.
- Identify the actual value from the problem: $Actual\ y = 19$.
- Calculate the difference: $Actual - Predicted = 19 - 17 = 2$.
- Answer: $2$.
Example 3: Hard (Shifting the Axis / Abstract Constants)
Question: The line of best fit for a set of data is given by the equation $y = -4.2x + 150$, where $x$ represents the number of years since 2000. If a new line of best fit is created where $t$ represents the number of years since 2010, the new equation is $y = mt + k$. What is the value of $k$?
SAT Strategy: Shifting the starting year changes the $y$-intercept but keeps the slope identical. You need to find the value of the original model at the new "year zero" (which is 2010).
- The new variable $t$ is years since 2010. The intercept $k$ occurs when $t = 0$.
- In the context of the original equation, the year 2010 corresponds to $x = 10$.
- Substitute $x = 10$ into the original equation to find the new intercept: $y = -4.2(10) + 150$.
- $y = -42 + 150 = 108$.
- Answer: $108$.
Practice Questions
Level 1: Easy (Foundational SAT)
- The line of best fit for a scatterplot is given by the equation $y = 8.4x + 12$. What is the predicted value of $y$ when $x = 5$? (Grid-in)
- A mechanic tracks the number of hours, $h$, it takes to repair an engine based on the engine's age, $a$, in years. The line of best fit is $h = 1.2a + 2.5$. What is the best interpretation of $1.2$ in this context?
- A) The predicted repair time for a brand-new engine.
- B) The estimated increase in repair time, in hours, for each additional year of an engine's age.
- C) The estimated increase in an engine's age for each additional hour of repair time.
- D) The total number of hours required to repair an engine.
- A scatterplot shows the relationship between the outside temperature and the number of cups of hot chocolate sold at a cafe. The data shows a strong negative linear association. Which of the following values could be the correlation coefficient, $r$, for this data?
- A) $-0.92$
- B) $-0.15$
- C) $0.15$
- D) $0.92$
- A line of best fit is given by $y = -3x + 40$. For a given data point, the actual $x$-value is 8 and the actual $y$-value is 18. Does the line of best fit overpredict or underpredict the $y$-value for this point, and by how much?
- A) It overpredicts by 2.
- B) It underpredicts by 2.
- C) It overpredicts by 16.
- D) It underpredicts by 16.
- A scatterplot exhibits an exponential trend where the $y$-values decrease rapidly at first and then gradually level out as $x$ increases. Which type of model would best fit this data?
- A) Decreasing linear
- B) Increasing linear
- C) Exponential growth
- D) Exponential decay
Level 2: Medium (Standard SAT)
- A scatterplot contains 15 data points. The line of best fit is calculated and drawn on the plot. For exactly 9 of the data points, the actual $y$-value is strictly less than the predicted $y$-value. How many data points lie on or above the line of best fit? (Grid-in)
- The table below summarizes data collected on the length of a metal spring, $L$, in millimeters, when a weight of $w$ grams is attached to it.
{|c|c|c|c|c|}
$w$ (grams) & $10$ & $20$ & $30$ & $40$
$L$ (mm) & $45$ & $55$ & $65$ & $75$
Which of the following equations best models the relationship between $w$ and $L$?
- A) $L = 10w + 35$
- B) $L = w + 35$
- C) $L = 10w + 45$
- D) $L = 1.1w^2 + 35$
- The line of best fit for a set of data is $y = 0.8x - 12$. A data point has an actual $y$-value of $20$. If the line of best fit overpredicts the $y$-value for this data point by exactly $4$, what is the $x$-value of the data point? (Grid-in)
- An exponential model of the form $y = a(b)^x$ is fit to a scatterplot. The parameters are found to be $a = 150$ and $b = 1.08$. What is the predicted percentage increase in $y$ for every increase of $1$ in $x$? (Grid-in)
- A researcher creates a scatterplot comparing the population of a city (in thousands), $y$, to the number of years since 1990, $x$. The line of best fit is $y = -2.4x + 85$. According to the model, in what year is the population predicted to reach 25 thousand?
- A) 2005
- B) 2010
- C) 2015
- D) 2025
Level 3: Hard (Advanced SAT)
- The line of best fit for a data set relating $x$ and $y$ is given by $y = mx + c$, where $m > 0$ and $c < 0$. Which of the following must be true about the scatterplot of the data?
- A) The data shows a negative correlation.
- B) As $x$ increases, $y$ tends to increase, and the line intersects the $y$-axis below the origin.
- C) As $x$ increases, $y$ tends to decrease, and the line intersects the $y$-axis above the origin.
- D) All data points in the scatterplot must have negative $y$-values.
- The number of active users on a platform, $U$, in millions, $t$ years after its launch is modeled by the exponential function $U(t) = 4(1.5)^t$. A linear model is also created for the same data during the first 2 years, resulting in the line of best fit $U_L(t) = 2.5t + 4$. What is the difference, in millions, between the predictions made by the exponential model and the linear model when $t = 2$? (Grid-in)
- A real estate agent modeled the price of houses, $P$, in dollars, based on their distance from the city center, $d$, in miles, using the equation $P = -15,000d + 450,000$. The agent later decides to rewrite the model so that distance is measured in kilometers, $k$. If 1 mile is approximately 1.6 kilometers, which equation best represents the rewritten model?
- A) $P = -24,000k + 450,000$
- B) $P = -9,375k + 450,000$
- C) $P = -15,000k + 720,000$
- D) $P = -15,000k + 281,250$
- A data set has a line of best fit given by $y = 5x + 10$. An extreme outlier is removed from the data set, and a new line of best fit is calculated as $y = 2x + 10$. Which of the following best describes the outlier that was removed?
- A) A point with a large positive $x$-value and a very large positive $y$-value.
- B) A point with a large positive $x$-value and a very small negative $y$-value.
- C) A point with an $x$-value of 0 and a very large positive $y$-value.
- D) A point with an $x$-value of 0 and a very small negative $y$-value.
- A scatterplot has a line of best fit with the equation $y = 4.5x - 10$. If every $x$-value in the original data set is multiplied by 2 and every $y$-value remains the same, what will be the slope of the new line of best fit? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- A linear regression model $y = mx + c$ was fitted to a set of data points predicting a company's revenue, $y$, in millions, $x$ years after 2015. The model predicted revenues of $14$ million in 2018 and $26$ million in 2022. What is the predicted revenue, in millions, in 2026? (Grid-in)
- A scatterplot shows a non-linear relationship between $x$ and $y$. However, when the researcher plots $x$ against the natural logarithm of $y$ ($\ln(y)$), the data forms a perfect straight line modeled by the equation $\ln(y) = 3x + 2$. Which of the following equations models the original relationship between $x$ and $y$?
- A) $y = e^{3x} + e^2$
- B) $y = e^2(e^3)^x$
- C) $y = 3e^x + 2$
- D) $y = 2(3)^x$
- Model A predicts the population of Town A, in thousands, $t$ years after 2000 with the equation $P_A = 12t + 45$. Model B predicts the population of Town B with $P_B = 18t + 15$. According to the models, in what year will the population of Town B exceed the population of Town A by exactly 21 thousand? (Grid-in)
- The line of best fit for a set of data is $y = 3x + 5$. There are three data points: $(2, a)$, $(4, b)$, and $(6, c)$. If the sum of the residuals (the differences between the actual $y$-values and the predicted $y$-values) for these three points is exactly $0$, what is the value of $a + b + c$? (Grid-in)
- A researcher models the relationship between $x$ and $y$ using the exponential function $y = p(q)^x$. The data shows that when $x$ increases by $2$, the predicted value of $y$ increases by $44%$. If $q > 0$, what is the value of $q$? (Round to the nearest hundredth). (Grid-in)
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