Question

LaTasha was presented with the following data set and argued that there was no correlation between x and y. Is LaTasha correct? Use the regression equation to explain your reasoning.

A 2-row table with 7 columns. The first row is labeled x with entries 1, 2, 3, 4, 5, 6, 7. The second row is labeled y with entries 4, 5, 4, 5, 4, 5, 4.

Answer 1

Compute the least-squares line y = a + bx.

x̄ = (1+2+3+4+5+6+7)/7 = 4
ȳ = (4+5+4+5+4+5+4)/7 = 31/7 ≈ 4.4286

Slope b = Σ(xi−x̄)(yi−ȳ) / Σ(xi−x̄)^2.
Compute the numerator:
Σ(xi−4)(yi−31/7) = (−3)(−3/7)+(−2)(4/7)+(−1)(−3/7)+0(4/7)+1(−3/7)+2(4/7)+3(−3/7) = 0.

Thus b = 0 and a = ȳ − b x̄ = 31/7. The regression line is
y = 31/7 ≈ 4.4286 (a horizontal line).

Because the slope is zero, there is no linear correlation between x and y. (Note: the y-values do follow a non‑linear alternating pattern, but that does not produce a linear association.) LaTasha is correct if she meant no linear correlation.