Which pair of normal equations gives $a$ and $b$ in the least-squares line $y = a + bx$ for $n$ observations?

\sum y = a + b\sum x

and

\sum xy = a\sum x + b\sum x^{2}

Correct answerYour answer

\sum y = na + b\sum x

and

\sum xy = na + b\sum x^{2}

Correct answerYour answer

\sum x = na + b\sum y

and

\sum xy = a\sum y + b\sum y^{2}

Correct answerYour answer

\sum y = na + b\sum x

and

\sum xy = a\sum x + b\sum x^{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Summing $y = a + bx$ over all $n$ points gives $\sum y = na + b\sum x$ .
Multiplying by $x$ first and then summing gives $\sum xy = a\sum x + b\sum x^{2}$ .

Worth noting

The second pair writes $\sum y = a + b\sum x$ : adding $a$ up $n$ times gives $na$ , not $a$ .

Ten questions from Time Series & Trend, at your own pace.

Which pair of normal equations gives aaa and bbb in the least-squares line y=a+bxy = a + bxy=a+bx for nnn observations?