10.4 Judging Whether a Normal Model Fits and Choosing a Regression Model

10.4 Judging Whether a Normal Model Fits and Choosing a Regression Model

[!NOTE] Regents Standard Alignment: NYSED split the old S-ID.4 into two assessed pieces for Algebra II. AII-S.ID.4a - recognize whether or not a normal curve is appropriate for a given data set. AII-S.ID.4b - if appropriate, determine population percentages using a graphing calculator for an appropriate normal curve. Section 10.1 handles 4b. This section handles 4a, plus the model-selection half of AII-S.ID.6a, where NYSED states that "when fitting a function to real-world data, emphasis is on quadratic, exponential and power models, including the regression capabilities of the calculator."

The word if in standard 4b carries real weight. NYSED deliberately made "should I use a normal model at all?" a question you can be asked on its own, without any percentage to compute.

---

1. The Normality Checklist

Run these five checks against a histogram, dot plot, or box plot before you touch normalcdf.

[!IMPORTANT] Skew is detected by which tail is long, and it is named for that tail. A distribution with a long right tail is right-skewed, and its mean exceeds its median. If a prompt reports a mean of $$74{,}000$ and a median of $$52{,}000$, you already know the distribution is right-skewed and a normal model is inappropriate, before you see a single graph.

Why It Matters Numerically

Applying the Empirical Rule to skewed data produces claims that are simply false. Suppose annual household incomes have mean $$74{,}000$ and standard deviation $$45{,}000$. A normal model would assert that about $16%$ of households fall below $\mu - \sigma = $29{,}000$ and, symmetrically, that about $2.5%$ fall below $\mu - 2\sigma = -$16{,}000$ - a negative income. The impossibility of that second figure is the tell: the model does not fit the data.

---

2. Worked Judgment 1: A Distribution That Fits

Problem: A quality engineer measures the diameters of 200 machined bearings. The dot plot is single-peaked and roughly symmetric about 12.00 mm, with values thinning out evenly toward 11.94 mm and 12.06 mm and no isolated points. The sample mean is 12.001 mm and the sample median is 12.000 mm. Is a normal model appropriate? Justify.

• Shape and modality: single peak, no secondary cluster.
• Symmetry: the tails thin out at comparable rates in both directions.
• Gaps and outliers: none reported.
• Center agreement: mean $12.001 \approx$ median $12.000$.
• Conclusion: Yes. The distribution is approximately symmetric and single-peaked with no gaps or outliers, and the mean and median agree to three decimal places, so a normal curve is an appropriate model and Empirical Rule or normalcdf calculations may be used.

That last sentence is the form a justification should take: name the evidence, then state the verdict.

---

3. Worked Judgment 2: Two Distributions That Do Not Fit

Problem A - skew. The number of minutes 150 customers waited in a pharmacy queue has a dot plot with a tall cluster between 0 and 5 minutes and a long thin tail extending out to 42 minutes. Mean $= 8.4$, median $= 4.5$.

• The long right tail and the mean sitting well above the median identify strong right skew. A normal model is not appropriate. Reporting "about 68% of waits fall within one standard deviation of the mean" would badly misdescribe a distribution where most waits cluster near zero.

Problem B - bimodality. The heights of all 400 students at a school produce a dot plot with two distinct peaks, one near 64 inches and one near 70 inches, with a visible dip between them.

• Two peaks mean two overlaid subpopulations. A normal model is not appropriate for the combined data set, although a normal model might well fit each subgroup separately. The correct response names the two modes as the reason.

---

4. Choosing a Regression Model (AII-S.ID.6a)

Once a scatterplot shows an association, the second judgment is which family to fit. Work from the data pattern, not from habit.

Fingerprinting an Equally Spaced Table

Worked Selection A - quadratic

$x$: $1, 2, 3, 4, 5$ and $y$: $3, 12, 27, 48, 75$.

• First differences: $9, 15, 21, 27$ - not constant, so not linear.
• Second differences: $6, 6, 6$ - constant, so quadratic. Fitting with QuadReg recovers $y = 3x^2$.

Worked Selection B - exponential

$x$: $0, 1, 2, 3, 4$ and $y$: $5, 15, 45, 135, 405$.

• Ratios: $\frac{15}{5} = 3$, $\frac{45}{15} = 3$, $\frac{135}{45} = 3$, $\frac{405}{135} = 3$ - constant, so exponential. ExpReg recovers $y = 5(3)^x$.

Worked Selection C - power

$x$: $1, 4, 9, 16$ and $y$: $2, 4, 6, 8$.

• The inputs are not equally spaced, so differences are not the right tool. Look instead at what happens when the input is multiplied: going from $x = 1$ to $x = 4$ multiplies $x$ by $4$ and multiplies $y$ by $2$; going from $x = 4$ to $x = 16$ again multiplies $x$ by $4$ and again multiplies $y$ by $2$.
• A constant output ratio per constant input multiplier is the signature of a power model. Since multiplying $x$ by $4$ multiplies $y$ by $4^{,b} = 2$, we get $b = \tfrac{1}{2}$, and PwrReg recovers $y = 2x^{1/2} = 2\sqrt{x}$.

[!TIP] Power versus exponential is the hard call, because both curve upward. The clean discriminator is what stays constant. An exponential multiplies $y$ by a fixed factor for each fixed addition to $x$. A power model multiplies $y$ by a fixed factor for each fixed multiplication of $x$. A second practical difference: a power model with $b > 0$ passes through the origin, while an exponential with $a > 0$ never reaches zero.

---

5. Reporting a Model-Selection Decision

A full-credit response names the evidence and then commits.

• Weak: "It looks exponential."
• Full credit: "The ratios of consecutive outputs are 3, 3, 3, and 3, so the data change by a constant factor over equal intervals. An exponential model is appropriate, and ExpReg gives $y = 5(3)^x$."

The same discipline applies to normality:

• Weak: "The data are normal."
• Full credit: "The dot plot is single-peaked and roughly symmetric with no gaps or outliers, and the mean and median are nearly equal, so a normal curve is an appropriate model for this data set."

Finally, remember NYSED's caution attached to this domain: an association between two variables never by itself establishes that one causes the other to change, and the two variables must be measured on the same subjects rather than being two unrelated quantitative lists.