Compute the least squares line from summary statistics

Which calculation should be completed first?

Predict how the slope changes when correlation changes sign while both means and spreads stay fixed. Move the correlation setting, including zero, and inspect the mean point

What remains true for every displayed fit?

Least-squares line from summary statistics

With x as the explanatory variable and y as the response variable, the slope uses the correlation and both spreads. Find b before a, and keep unrounded values until the final reporting step

1. 1Assign the roles Use x for the explanatory variable and y for the response variable \(\displaystyle x=\text{EV},\quad y=\text{RV}\)
2. 2Calculate the slope Multiply r by the response standard deviation divided by the explanatory standard deviation \(\displaystyle b=(-0.8)\frac{15}{10}=-1.2\)
3. 3Calculate the intercept Subtract b times the explanatory mean from the response mean \(\displaystyle a=42-(-1.2)(10)=54\)
4. 4Round at the end Use the exact slope when finding the intercept, then round the reported coefficients as requested \(\displaystyle \widehat{y}=54-1.2x\)

Given values

\(\displaystyle r=-0.8,\ s_x=10,\ s_y=15\)

Use the summary values to calculate the slope b.

Given values

\(\displaystyle \bar{x}=10,\ \bar{y}=42,\ b=-1.2\)

Use b to calculate the intercept a without rounding b first

Given values

\(\displaystyle r=-0.8, s_x=10, s_y=15\)

Reversing the spread ratio. One of these lines is right. Which one?

Produce a complete equation with stated precision

A fictional sensor investigation uses elapsed time x in seconds to explain output y in sensor units. Treat the supplied summaries as exact. Calculate b and a without intermediate rounding. Write the full prediction equation and then report both coefficients to four significant figures. Also report a to three significant figures, using scientific notation to make the precision explicit

1. 1Verify the calculation Record a calculator or spreadsheet expression for each coefficient, preserving the stored slope for the intercept. Check that the unrounded line passes through (400,10000), and recover r from b and both spreads. State the units of the slope
2. 2Save your work before opening the model Keep the calculations, complete equation, interpretation and tool record

Model response: significant figures count from the first nonzero digit

b=(−0.716)×200/125=−1.1456 sensor units per second. a=10000−(−1.1456)×400=10458.24 sensor units. The unrounded prediction is ŷ=10458.24−1.1456x. At four significant figures, report ŷ≈1.046×10⁴−1.146x. The intercept alone at three significant figures is 1.05×10⁴. The rounded equation is a report; retain the unrounded coefficients for later calculations

1. 1Distinguish the precision instruction Four significant figures in 10458.24 gives 10460, whereas four decimal places gives 10458.2400. Scientific notation 1.046×10⁴ makes all four significant figures explicit. For the slope, four significant figures gives−1.146, whereas four decimal places gives−1.1456
2. 2Independent checks and one workflow Store r, x mean, x spread, y mean and y spread in labelled cells. Calculate b=r*s_y/s_x, then a=y_mean−b*x_mean, referring to the unrounded b cell. Check that the unrounded line passes through the mean point and b×125/200=−0.716. Do not use a rounded display value as a new input

Given values

\(\displaystyle r=-0.716,\ s_x=125,\ s_y=200\)

Calculate the sensor model’s unrounded slope

Given values

\(\displaystyle \bar{y}=10000,\ \bar{x}=400,\ b=-1.1456\)

Calculate the intercept using the unrounded slope

Given values

\(\displaystyle b=-1.1456\)

Report b to four significant figures, using a decimal

Given values

\(\displaystyle a=10458.24\)

Report a to four significant figures in scientific notation, for example using e notation

Given values

\(\displaystyle a=10458.24\)

Report a to three significant figures in scientific notation

Given values

\(\displaystyle b=-1.1456,\ s_x=125,\ s_y=200\)

Recover r from the unrounded slope and spreads

Why retain unrounded coefficients after reporting a rounded equation?