Fit both regression directions and recover the means

For a small dataset the source gives a home-screen alternative to STAT Edit: enter {16,13,16,15,16,12,17,15} STO→ L1 and {7,17,12,11,18,10,17,12} STO→ L2. Use curly braces, comma-separated values, STO→, then 2nd 1 for L1 or 2nd 2 for L2. Preserve the paired order.

Press STAT → CALC → 8:LinReg(a+bx). Set Xlist=L1, Ylist=L2. Enter Y1 in Store RegEQ using ALPHA TRACE 1, highlight Calculate, then ENTER. Y1 retains this fit for the sketch. The display gives $a=7.75$, $b=0.35$, $r^2=0.0226851852$ and $r=0.150616019$. Here the calculator’s x and y match the question variables: $a$ is the intercept and $b$ the gradient.

Independently, $S_{xx}=20$ and $S_{xy}=7$ give $b=7/20$ and $a=13-(7/20)15=7.75$.

Again select STAT → CALC → 8:LinReg(a+bx), but set Xlist=L2, Ylist=L1, highlight Calculate, then ENTER. The calculator’s displayed x now represents the question’s $y$ and its displayed y represents the question’s $x$. Write the result with the original variables in their correct roles.

The printed output is $a=14.15740741$, $b=0.0648148148$, with the same $r^2$ and $r$. Hence $x=0.0648148148y+14.15740741$, or the source’s 3-s.f. form $x=0.0648y+14.2$. Independently, $d=S_{xy}/S_{yy}=7/108$ and $c=15-(7/108)13=1529/108$. Keep the exact or full-precision values for later calculations; do not invert the first fitted line.

1. Press 2nd Y= to open STAT PLOTS.
2. Press ENTER for Plot1; select ON, scatter type, Xlist=L1, Ylist=L2 (the source uses a square marker and blue colour).
3. Press Y=. Y1 already contains $0.35X+7.75$. To plot the reverse fit on these same axes, make $y$ the subject and enter it in Y2: $Y2=(X-c)/d$. The source screen uses $(X-14.15741)/0.064810$; use stored full coefficients or $Y2=13+(108/7)(X-15)$ to avoid rounding effects.
4. Press ZOOM → 9:ZoomStat to include all observations and both fits.
5. Press TRACE, using left/right arrows to see each observation’s coordinates at the screen bottom.

The printed sketch labels observed extrema $x=12,17$ and $y=7,18$, with a cropped lower-left O; this does not mean the plotted corner is the true origin. Its blue caption prints $y=0.35+7.75$, omitting $x$. The earlier equation and calculator screen correctly show $y=0.35x+7.75$; the reconstruction uses that equation. Its reverse-line caption is rounded to $x=0.0648y+14.2$, while this plot uses exact coefficients. All eight points are preserved.

Both lines pass through $(\bar x,\bar y)$. Solve their full-precision equations together:

$y=\frac7{20}x+\frac{31}4,\qquad x=\frac7{108}y+\frac{1529}{108}.$

Substitution gives $x=15$, then $y=13$. The source’s calculator intersection procedure is 2nd TRACE → 5:intersect, then ENTER for the first curve, ENTER for the second curve and ENTER for the guess. Its display is approximately $(14.999999,12.999999)$, reported as $(15.0,13.0)$ to 3 s.f. Small display errors come from stored rounding, not different means.

The unnumbered final ask specifically requires the regression lines. The source offers two checks when that restriction does not apply: calculate the arithmetic means directly, $\bar x=120/8=15$ and $\bar y=104/8=13$; or use STAT → CALC → 2:2-Var Stats, Xlist=L1, Ylist=L2, Calculate → ENTER, then scroll through the statistics. The printed totals are $\sum x=120$, $\sum x^2=1820$, $\sum y=104$, $\sum y^2=1460$, $\sum xy=1567$, $n=8$, with sample standard deviations $S_x=1.690308509$, $S_y=3.927922024$ and population standard deviations $\sigma_x=1.58113883$, $\sigma_y=3.674234614$. These independently recover the same means.