10.2 Bivariate Data, Scatterplots, and Regression Models

10.2 Bivariate Data, Scatterplots, and Regression Models

Quick Answer: Bivariate quantitative data pairs two numerical variables to explore relationships between an explanatory variable ($x$) and a response variable ($y$). By examining scatterplots, students classify trends into four core function models: linear ($y = ax + b$), quadratic ($y = ax^2 + bx + c$), exponential ($y = a \cdot b^x$), and sinusoidal ($y = a\sin(bx + c) + d$). On graphing calculators, regression models are calculated under STAT $\to$ CALC. Model parameters represent contextual metrics such as initial states, unit rates of change, multiplicative growth/decay factors, and periodic amplitudes. Predictions must use unrounded coefficients to prevent cumulative rounding errors.

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1. Bivariate Quantitative Data: Explanatory vs. Response Variables (AII-S.ID.6a)

[!NOTE] Scope note. The assessed component is AII-S.ID.6a - fit a function to real-world data. NYSED states the emphasis directly: "When fitting a function to real-world data, emphasis is on quadratic, exponential and power models, including the regression capabilities of the calculator." Power regression is named explicitly, and the Educator Guide sample item for this standard is a power-regression question. NYSED also warns that bivariate data must be linked to the same subjects and that an association never by itself implies that one variable causes the other to change.

In univariate statistics, analysts examine single variables in isolation (such as mean test score or standard deviation of heights). In contrast, bivariate data involves pairs of numerical measurements, denoted $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$, collected from the same individual or experimental unit to investigate whether changes in one variable correspond to changes in another.

Distinguishing Variable Roles

• Explanatory Variable ($x$): Also called the independent variable or predictor. It represents the input variable that is hypothesized to influence, explain, or predict changes in the second variable. Plotted strictly on the horizontal axis.
• Response Variable ($y$): Also called the dependent variable or outcome. It represents the measurement or result that responds to variations in the explanatory variable. Plotted strictly on the vertical axis.

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2. Scatterplot Patterns and Function Families

A scatterplot displays bivariate data points as coordinates on the Cartesian plane. The visual arrangement of points reveals the mathematical family best suited to model the relationship.

Linear (Steady Change)        Quadratic (Parabolic Arc)
   y ^    *                      y ^       * *
     |   *                         |     *     *
     |  *                          |    *       *
     | *                           |   *         *
     +---------> x                 +---------------> x

Exponential (Rapid Growth)    Sinusoidal (Periodic Wave)
   y ^        *                  y ^    *       *
     |       *                     |   * *     * *
     |     *                       |  *   *   *   *
     | * *                         | *     * *     *
     +---------> x                 +-----------------> x

The Four Regression Function Families

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3. Graphing Calculator Regression Commands (STAT CALC)

The Regents examination requires candidates to generate least-squares regression equations using a standard graphing calculator.

Calculator Operational Workflow

1. Enter Data: Press STAT $\to$ 1:Edit.... Enter explanatory values ($x$) into list L1 and response values ($y$) into list L2.
2. Execute Regression: Press STAT, scroll right to CALC, and select the required regression model:
   • 4:LinReg(ax+b) for linear regression.
   • 5:QuadReg for quadratic regression.
   • 0:ExpReg for exponential regression ($y = a \cdot b^x$).
   • A:PwrReg for power regression ($y = a \cdot x^{,b}$).
   • C:SinReg for sinusoidal regression ($y = a\sin(bx + c) + d$).
3. Store Regression Equation: In the regression setup screen, navigate to Store RegEQ:. Press VARS $\to$ Y-VARS $\to$ 1:Function $\to$ 1:Y1 $\to$ Calculate. This stores the full, unrounded model directly into function Y1.

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4. Contextual Interpretation of Model Parameters

Regents free-response questions frequently ask students to explain the physical meaning of regression parameters in context.

Linear Models: $y = ax + b$

• Slope ($a$): The average predicted change in the response variable $y$ for each one-unit increase in the explanatory variable $x$. Always include units: $\frac{\text{units of } y}{\text{units of } x}$.
• $y$-Intercept ($b$): The baseline value of the response variable $y$ when the explanatory variable $x = 0$.

Exponential Models: $y = a \cdot b^x$

• Initial Value ($a$): The starting amount of the response variable at $x = 0$.
• Base / Growth Factor ($b$): The multiplier applied to $y$ for each one-unit increase in $x$.
  • If $b > 1$, the model represents exponential growth with percentage rate $r = b - 1$.
  • If $0 < b < 1$, the model represents exponential decay with percentage rate $r = 1 - b$.

Power Models: $y = a \cdot x^{,b}$

• Coefficient ($a$): The predicted response when $x = 1$, since $a \cdot 1^{,b} = a$. It is not an initial value at $x = 0$ - for $b > 0$ a power model passes through the origin, and for $b < 0$ it is undefined there.
• Exponent ($b$): The scaling exponent. Multiplying the explanatory variable by a factor $k$ multiplies the predicted response by $k^{,b}$. If $b = 2$, doubling $x$ quadruples $y$; if $b = 0.5$, quadrupling $x$ doubles $y$.
• Telling power apart from exponential: Both curve upward, and both can fit the same points reasonably well. The distinguishing test is the behavior of the ratios. An exponential model multiplies $y$ by a constant factor for each equal-sized increase in $x$; a power model multiplies $y$ by a constant factor for each equal-sized multiplication of $x$. On the calculator, run both ExpReg and PwrReg on the same lists and compare the fit against the residual pattern before committing.
• Worked check: For the bacterial data $(1, 2), (3, 25), (5, 81), (7, 175), (9, 310), (11, 497)$ used in NYSED sample material, PwrReg returns approximately $y = 2.001x^{2.298}$. Notice that $x$ tripling from 1 to 3 multiplies $y$ by about 12.5, while $x$ tripling again from 3 to 9 also multiplies $y$ by about 12.4 - the constant-ratio-per-multiplication signature of a power model.

Sinusoidal Models: $y = a\sin(bx + c) + d$

• Amplitude ($|a|$): The maximum displacement from the midline: $|a| = \frac{\text{Maximum } y - \text{Minimum } y}{2}$.
• Midline / Vertical Shift ($d$): The average or baseline equilibrium level: $d = \frac{\text{Maximum } y + \text{Minimum } y}{2}$.
• Frequency Parameter ($b$): Governs the cycle length. The period $T$ is given by $T = \frac{2\pi}{b}$, which implies $b = \frac{2\pi}{T}$.

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5. Predictions, Interpolation, Extrapolation, and Rounding Protocols

Once a regression equation is determined, it is utilized to predict unknown response values from specified explanatory inputs.

Interpolation vs. Extrapolation

• Interpolation: Predicting values within the domain range of the original sample data. These predictions are generally reliable because the mathematical model has been validated across this interval.
• Extrapolation: Predicting values outside the observed data domain. Extrapolations are inherently hazardous because underlying physical, economic, or biological trends often alter beyond observed bounds.

[!CAUTION] The Cardinal Rule Against Premature Rounding: Never substitute rounded intermediate coefficients back into an equation to calculate a prediction. Rounding coefficients such as slope or exponential base causes significant error magnification. Always evaluate the full unrounded equation on the calculator (Y1(value)), and round only the final output to the specified precision.

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6. Worked Examples

Worked Problem 1: Exponential Regression and Parameter Interpretation

Problem: A laboratory technician records the concentration $C(t)$ of an antibiotic in milligrams per liter over time $t$ in hours. The calculator yields $C(t) = 48.214(0.817)^t$. Round parameters to the nearest hundredth, state the complete model, and interpret the base $b$ in context.

• Step 1: Round coefficients to the nearest hundredth. $a \approx 48.21, \quad b \approx 0.82$ The rounded regression equation is: $C(t) = 48.21(0.82)^t$
• Step 2: Interpret the initial value $a$. At time $t = 0$ hours, the initial antibiotic concentration in the patient's bloodstream was approximately $48.21\text{ mg/L}$.
• Step 3: Interpret the base $b$. Because $b = 0.82 < 1$, the drug undergoes exponential decay. The decay rate is: $r = 1 - b = 1 - 0.82 = 0.18 = 18\%$ Each hour, the antibiotic concentration decreases by approximately $18%$ of its value from the preceding hour.

Worked Problem 2: Sinusoidal Regression Modeling

Problem: Coastal water levels oscillate with the tides. High tide reaches a maximum depth of $14.6\text{ ft}$, and low tide reaches a minimum depth of $3.2\text{ ft}$. The time elapsed between two consecutive high tides is exactly $12.4\text{ hours}$. Construct a sinusoidal model in the form $y = a\cos(bt) + d$ assuming high tide occurs at $t = 0$.

• Step 1: Calculate the amplitude $a$. $a = \frac{\text{Max} - \text{Min}}{2} = \frac{14.6 - 3.2}{2} = \frac{11.4}{2} = 5.7\text{ ft}$
• Step 2: Calculate the vertical shift (midline) $d$. $d = \frac{\text{Max} + \text{Min}}{2} = \frac{14.6 + 3.2}{2} = \frac{17.8}{2} = 8.9\text{ ft}$
• Step 3: Calculate the frequency parameter $b$. The period is $T = 12.4\text{ hours}$. Using the period formula $T = \frac{2\pi}{b}$: $b = \frac{2\pi}{12.4} = \frac{\pi}{6.2} \approx 0.5067$
• Step 4: Assemble the regression equation. $y = 5.7\cos(0.51t) + 8.9$

Worked Problem 3: Preventing Premature Rounding During Predictions

Problem: A researcher computes the linear regression model $y = 0.1487x + 12.314$ relating vehicle weight $x$ (in hundreds of pounds) to stopping distance $y$ (in feet). Predict the stopping distance for a vehicle weighing $3800\text{ lb}$ ($x = 38$) to the nearest tenth of a foot, and show why premature rounding to $y = 0.15x + 12.3$ is penalized.

• Step 1: Evaluate using full precision. $y = 0.1487(38) + 12.314 = 5.6506 + 12.314 = 17.9646 \approx 18.0\text{ ft}$
• Step 2: Evaluate using prematurely rounded coefficients. $y_{\text{rounded}} = 0.15(38) + 12.3 = 5.70 + 12.3 = 18.0\text{ ft} \quad (\text{matches here coincidentally})$ However, for $x = 65$:
  • Full precision: $0.1487(65) + 12.314 = 9.6655 + 12.314 = 21.9795 \approx 22.0\text{ ft}$.
  • Prematurely rounded: $0.15(65) + 12.3 = 9.75 + 12.3 = 22.05 \approx 22.1\text{ ft}$. The rounding error shifts the tenths digit, resulting in lost credit on the Regents exam.

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7. Common Regents Pitfalls & Exam Strategies

• Pitfall 1: Confusing Multiplicative Base with Percentage Rate. In $y = 250(1.06)^x$, the annual growth rate is $6%$, not $106%$. Conversely, in $y = 250(0.94)^x$, the decay rate is $6%$, not $94%$.
• Pitfall 2: Transposing Lists in the Calculator. Entering response values into L1 and explanatory values into L2 inverts the slope and yields an entirely invalid model.
• Pitfall 3: Failing to Match the Specified Rounding Precision. Regents instructions specify rounding places for each parameter (e.g., "round $a$ and $b$ to the nearest thousandth"). Truncating digits forfeits credit.