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Material Type: Notes; Class: Intro Stat Inference; Subject: Mathematics; University: University of Utah; Term: Spring 2003;
Typology: Study notes
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SD/Correlation Computations in Easy Steps Math 1070-1, Spring 2003 (Univ. of Utah)
Example: (SL-County unemployment data)
Year (x) 1997 1998 1999 2000 2001 Rate in % (y) 2.7 3.4 3.4 3 4.
Sample size: n = 5.
Means: ¯x = (1997 + · · · + 2001)/ 5 = 1999, ¯y = (2.7 + · · · + 4.3)/ 5 = 3.36.
The standard Deviation of x: (First the square)
S x^2 =
n − 1
∑ (x − x¯)^2
What does this mean? (^) (xx −^ − ¯x¯x)^2 −^24 −^11 00 11
So: S x^2 = (^14) (4 + 1 + 0 + 1 + 4) = 2. 5. There- fore, S (^) x =
Also, (^) (yy −− y^ ¯¯y)^2 −^00 ..^6644 0.^040 0.^040 −^00 ..^3613 00 ..^9488
So:
S y^2 ≈
Therefore, S (^) y ≈
r = 1 n − 1
∑ (x − x¯ S (^) x
) (y − y¯ S (^) y
)
n − 1
∑ SUxSUy ,
where SU means “in standard units.” In other words, the above says, “first compute a col- umn of x in standard units and one for y. Then cross-multiply and add. Finally, divide by n − 1.” Now we are off to work out the details which I will take pains to do very meticulously so as to avoid those silly—and unacceptable— errors.
Regression The equation of the regression line is: SUy = rSUx. I.e.,
y − y¯ S (^) y
= r
(x − ¯x S (^) x
) .
Solve for y (DO IT!) to obtain:
y = rS (^) y
(x − x¯ S (^) x
)
=
(rS y S (^) x
)
︸ ︷︷ ︸ (slope)
x +
[ y ¯ −
(rS y S (^) x
) ¯x
]
︸ ︷︷ ︸ (intercept)
In our Example above, we had
x¯ = 1999 S (^) x ≈ 1. 58 y¯ = 3. 36 S (^) y ≈ 0. 6 r ≈ 0. 73.
So slope = (rS (^) y /S (^) x) ≈ (0. 73 × 0. 6 / 1 .58) = 0 .28 (without rounding). Similarly, intercept = − 556 .36 (without rounding; check this!) So, the regression line—in the previous Example— is:
y = 0. 28 x − 556. 36.
The regression-prediction for the unemployment in SL-county in the year x = 2001 (based on the above data):
y ≈ 0. 28 × 2001 − 556 .36 = 3.92%.
