Skip to main content

Fit straight line: humidity and moisture

Short answer

A straight line is reasonable because r = 0.9144, and the significance test t = 7.144 exceeds the critical value 2.228. The least squares line is Y = -0.9495 + 0.2690X, so each 1% humidity increase raises moisture content by about 0.269%.

Raw material used in the production of a synthetic fibre is stored in a place which has no humidity control. Measurements of the relative humidity in the storage place and moisture content of sample of the raw material (both in percentage) on 12 days yielded the following results:

Humidity, X423550434862313644395548
Moisture content, Y12814911167912101311
Verify that it is reasonable to fit a straight line. Fit the straight by the method of least squares.

Let X = relative humidity (%), Y = moisture content (%), n = 12.

Is a straight line reasonable?

Plotting the points (X, Y) gives a scatter that rises from lower-left to upper-right and lies close to a straight band, so a linear model is suitable. This is confirmed numerically by the correlation coefficient:

r = Sxy/√(Sxx·Syy) = 0.9144 (r² = 0.836)

Test: H₀: ρ = 0 against H₁: ρ ≠ 0, α = 0.05. t = r√(n−2)/√(1−r²) = 7.144, critical value t(0.025, 10) = 2.228. Since 7.144 > 2.228, H₀ is rejected: the linear relationship is significant.

Least squares line Y = a + bX

Sums (n = 12):

        x        y       xy       x²       y²
       42       12      504     1764      144
       35        8      280     1225       64
       50       14      700     2500      196
       43        9      387     1849       81
       48       11      528     2304      121
       62       16      992     3844      256
       31        7      217      961       49
       36        9      324     1296       81
       44       12      528     1936      144
       39       10      390     1521      100
       55       13      715     3025      169
       48       11      528     2304      121
---------------------------------------------
      533      132     6093    24529     1526

Σx = 533, Σy = 132, Σxy = 6093, Σx² = 24529, Σy² = 1526

Means: x̄ = 44.4167, ȳ = 11.0000

Sxx = Σx² − (Σx)²/n = 854.9167 Sxy = Σxy − (Σx)(Σy)/n = 230.0000 Syy = Σy² − (Σy)²/n = 74.0000

Slope b = Sxy/Sxx = 230.0000/854.9167 = 0.2690 Intercept a = ȳ − bx̄ = 11.0000 − (0.2690)(44.4167) = -0.9495

Fitted line: Ŷ = -0.9495 + 0.2690 X

Interpretation: each 1% rise in humidity raises moisture content by about 0.269%.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 5 questions →