ISS 2016 Statistics Paper-1 Solution: Question 29 (Regression Coefficient of Y on X)

Continuing our question-by-question walk through the ISS Statistics Paper-1 previous year papers, we now arrive at Question 29 of the 2016 paper. This one looks intimidating at first glance because it gives you two regression lines with an unknown constant a buried inside them, but once you know what a regression coefficient actually represents, the algebra becomes almost mechanical.

Quick Summary

• Topic: Regression lines of Y on X and X on Y, and the relationship between their slopes and the correlation coefficient

• Question Reference: ISS 2016, Statistics Paper-1, Question 29

• Correct Answer: Option (c) — 0.5 ≥ a ≥ 0, i.e. 0 ≤ a ≤ 0.5

The Question, Exactly As Asked

If Y = aX + 3 and X = 2Y + 6 are the regression lines of Y on X and X on Y respectively, then which one of the following is correct?

• (a) 0.5 ≤ a ≤ 1

• (b) a > 1

• (c) 0 ≤ a ≤ 0.5

• (d) −0.5 ≤ a ≤ 0

Step 1: Identify What Each Equation Actually Gives You

In regression, there are always two separate lines connecting the same two variables, and they are not the same line unless the correlation is perfect. The line "Y on X" is used to predict Y from a given value of X, and the line "X on Y" is used to predict X from a given value of Y. Each of these lines has its own slope, called a regression coefficient, and the two slopes are written as bYX (slope of Y on X) and bXY (slope of X on Y).

Here we are told:

• Regression line of Y on X: Y = aX + 3. Comparing this with the general form Y = bYXX + constant, we get bYX = a.

• Regression line of X on Y: X = 2Y + 6. Comparing this with the general form X = bXYY + constant, we get bXY = 2.

So the problem has really just handed us two numbers in disguise: bYX = a and bXY = 2. Everything else follows from the properties of these two coefficients.

Step 2: Recall the Key Identity Linking the Two Regression Coefficients

This is the fact the entire question is testing. For any pair of variables X and Y with correlation coefficient r, the two regression coefficients multiply to give the square of the correlation coefficient:

r² = bYX × bXY

This identity comes directly from how each coefficient is defined in terms of the covariance and the two standard deviations. If Cov(X, Y) = σXY, and the standard deviations of X and Y are σX and σY, then:

bYX = σXY / σX² and bXY = σXY / σY²

Multiplying these two together:

bYX × bXY = (σXY / σX²) × (σXY / σY²) = σXY² / (σX²σY²) = (σXY / (σXσY))² = r²

since r is defined as σXY / (σXσY). So the product of the two slopes is always exactly r², nothing more, nothing less.

Step 3: Use the Bound 0 ≤ r² ≤ 1

Because r itself always lies between −1 and +1 (that is one of the most basic facts about the correlation coefficient), its square r² can never be negative and can never exceed 1. In symbols:

0 ≤ r² ≤ 1

Substituting r² = bYX × bXY = a × 2 = 2a, this becomes:

0 ≤ 2a ≤ 1

Dividing every part of this inequality by 2:

0 ≤ a ≤ 0.5

Pro Tip: A very common slip here is to stop at "2a ≤ 1" and forget the lower bound "2a ≥ 0" entirely, which lets students carelessly pick an option like a > 1 out of habit. The way this is usually taught in a good classroom is to always write both halves of the double inequality for r² before touching the algebra, and to separately check the sign rule: bYX and bXY must always carry the

Step 4: Match with the Given Options and Confirm

We derived 0 ≤ a ≤ 0.5, which is exactly option (c). Checking it against the official ISS 2016 Statistics Paper-1 answer key confirms that the correct answer is option (c)— our independently worked-out result matches the official key exactly, so there is no discrepancy to flag for this question.

Why This Question Matters

Regression coefficients and their relationship to the correlation coefficient form one of the most heavily tested ideas across ISS, RBI Grade B, and other statistics-based competitive exams, precisely because a single short question like this one can test three things at once: whether you can read off a regression coefficient from a given line, whether you remember the identity r² = bYX × bXY, and whether you respect the basic bound 0 ≤ r² ≤ 1. Once this identity is internalised, entire categories of "find the range of the unknown constant" questions become two-line solutions instead of a source of anxiety.

Frequently Asked Questions

Why are there two separate regression lines instead of just one?

A regression line is built specifically to predict one variable from the other, and the method used to minimise prediction error is different depending on which variable is being predicted. Minimising the squared errors in Y gives one line (Y on X), while minimising the squared errors in X gives a different line (X on Y). The two lines coincide only when the correlation is exactly +1 or −1.

Can the two regression coefficients ever have opposite signs?

No. Since their product equals r², which can never be negative, bYX and bXY must always be of the same sign, or one (or both) of them must be zero. If you ever compute two regression coefficients with opposite signs from a given problem, it is a sign that an arithmetic error has crept in somewhere.

Is it possible for a regression coefficient to be greater than 1?

Yes, absolutely, a single regression coefficient can be any real number since it depends on the ratio of standard deviations as well as on r. It is only the product of the two regression coefficients, bYX × bXY, that is restricted to the range [0, 1], not each coefficient individually.

How do I quickly tell which line is "Y on X" and which is "X on Y" in a question?

Look at which variable is written alone on the left-hand side of the equals sign. If the equation is written as Y = (something with X), it is the regression line of Y on X, and the coefficient multiplying X is bYX. If X is alone on the left, it is the regression line of X on Y, and the coefficient multiplying Y is bXY.

What if the question had asked for the correlation coefficient r instead of just r²?

You would take the square root of bYX × bXY, but you would also need to determine the sign of r separately, since r² alone does not tell you whether r is positive or negative. The sign of r always matches the (common) sign of bYX and bXY.

Does this identity work for any two variables, or only for normally distributed data?

The identity r² = bYX × bXY follows purely from the algebraic definitions of covariance, variance, and correlation, so it holds for any two variables with finite variances, regardless of the underlying distribution.

If any step above felt shaky, especially the sign-consistency argument in Step 3, drop your doubt in the comments below and we will work through it together. And if you found this walkthrough useful, do share it with a fellow ISS aspirant who might be stuck on the same question.