Coding the middle year (2019) as $X=0$ makes $\sum X = 0$, which simplifies the two least-squares normal equations to direct formulas for $a$ and $b$.

Step 1 — Code the years. With 5 years, take the middle year (2019) as $X=0$, so 2017 = $-2$, 2018 = $-1$, 2019 = $0$, 2020 = $1$, 2021 = $2$. This makes $\sum X = -2-1+0+1+2 = 0$.

Step 2 — Build the working table:

Step 3 — Apply the simplified normal equations (valid because $\sum X = 0$):

So the fitted trend line is $Y = 24 + 3.6X$.

Step 4 — Read off the trend values by substituting each $X$:

Step 5 — Forecast 2022. 2022 codes to $X=3$: $Y = 24 + 3.6(3) = 24 + 10.8 = 34.8$.

Verification: substituting the normal-equation values back, $\sum Y$ should equal $na + b\sum X = 5(24)+3.6(0)=120$ ✓ (matches the table total), and $\sum XY$ should equal $a\sum X + b\sum X^2 = 24(0)+3.6(10)=36$ ✓ (matches the table total) — both check out, confirming $a=24, b=3.6$ are correct.

Alternative check: since $\sum X = 0$ was engineered by coding, $a$ always equals the simple arithmetic mean of $Y$ regardless of $b$ — here $(18+20+22+28+32)/5 = 120/5 = 24$, matching the value found from the normal equation, which is a quick independent check on $a$ alone (though not on $b$).

Coding the years starting from 1 instead of centring them on the middle year (which breaks $\sum X = 0$ and forces the full, harder pair of normal equations); mixing up which of $a$ and $b$ is the intercept and which is the slope; and forgetting to re-code the forecast year (2022 is $X=3$, not $X=5$) before substituting into the trend equation.