Question 4 OR (f)

Properties of regression coefficients; correlation from regression coefficients

4 marks

Time to write: About 8 minutes
Topic: Regression

Question

Question 4 OR (f) · 4 marks

State the properties of regression coefficients. Also, explain how correlation coefficients are computed from regression coefficients, with the help of an example.

Reworded from the university paper; the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

8 minfor 4 marks

Read from the question

State…Explain…2 asks, each with its own marks below. Answer both.
exampleExamples are asked for: one short example with each point.

Write it in this order

Read; underline the asks; jot your points 1 min
1 Properties of regression coefficients 2 marks 3 min
2 Method of finding r, with an example 2 marks 3 min
Check: every ask answered, key terms underlined 1 min

Short on time? Trim steps 1 and 2 evenly rather than skip one: each is capped at its own marks.

Try it first. About 8 minutes for 4 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Properties of regression coefficients. For the regression of Y on X the coefficient is byx = r σy / σx, and for X on Y it is bxy = r σx / σy.

Geometric mean property: r = ±√(byx × bxy); the correlation coefficient is the geometric mean of the two regression coefficients.
Same sign: both coefficients have the same sign, which is also the sign of r (both positive, or both negative).
Not both greater than one: since byx × bxy = r² ≤ 1, if one coefficient exceeds 1 (numerically) the other must be less than 1.
Arithmetic mean not less than r: (byx + bxy) / 2 ≥ r (for positive coefficients).
Independent of origin, not of scale: a change of origin leaves them unchanged; a change of scale alters them.
Not symmetric: byx and bxy are in general different, unlike r, which is the same both ways.

Correlation from the regression coefficients. Multiply the two coefficients, take the square root, and give the result the common sign of the coefficients: r = ±√(byx × bxy). When only the two regression equations are given, each is first written in the form Y = a + byx X and X = a′ + bxy Y; if the product of the coefficients so obtained exceeds 1, the equations have been assigned the wrong way round and must be swapped.

Example. Suppose the regression lines are 9X − 10Y + 20 = 0 and 5X − 2Y − 15 = 0.

Taking the first as Y on X: Y = 2 + 0.9X, so byx = 0.9. Taking the second as X on Y: X = 3 + 0.4Y, so bxy = 0.4.
byx × bxy = 0.9 × 0.4 = 0.36 ≤ 1, so the assignment is valid.
r = +√0.36 = 0.6 (positive, because both coefficients are positive).

(Had the product come out above 1, the lines would be re-assigned: the first as X on Y and the second as Y on X.)

Where the 4 marks sit

#	Component	Marks
1	Properties of regression coefficients	2
2	Method of finding r, with an example	2
	Total	4

Keyboard: ← → previous and next question, / search

Report a problemopens an email to admin@getkunji.com with the question ID872-Q4fORalready filled in.