৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

The correlation coefficient (r) ranges from –1 to +1. The closer |r| is to 1, the stronger the relationship. Here, r = –0.85 indicates a strong negative correlation.
Source: Applied General Statistics. Coxton and Crowden.

When r = 0, there is no linear correlation between the variables, but they may still have a non-linear relationship. Correlation would be called non linear if the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable. For example, if we double the amount of rainfall, the production of rice would not necessarily be doubled. It may be pointed out that in most practical cases we find a non linear relationship between the variables.

​Source: Business Statistics, SP Gupta, MP Gupta.

The coefficient of determination R² = r² × 100% = (0.9)² × 100% = 81%. This means that 81% of the variation in the dependent variable has been explained by the independent variable. The maximum value r² can take is 1. Not more than 1. Because r²=1 means all the variation is captured, not more than this variation can be achieved. 
Source: Business Statistics, SP Gupta, MP Gupta, Live MCQ class lecture.

Using r = Cov(X,Y)/(σ__X × σ__Y),  σ
 so, Cov(X,Y) = r×(σ_X × σ_Y)
   = 0.6 × 10 × 15 = 90.

Explanation: Spearman’s rank correlation uses ranked data, making it suitable for ordinal variables and capable of handling tied ranks.
Why Spearman’s ??
ρ: It’s a rank-based (non-parametric) measure—replace values with ranks and correlate the ranks. That makes it appropriate for ordinal data and more robust to outliers and certain nonlinear monotonic patterns. 
Why not Pearson’s ??
r: Pearson assumes roughly interval/ratio scale and linearity; using it on purely ordinal data can be misleading.
Source: Investopedia

Interpretation: In the simple linear model
Y = a + bX,
b is the regression coefficient (slope)—the expected change in Y for a one-unit increase in X, holding the model fixed. A negative b means an inverse relationship: as X rises, Y falls on average. 

Why η
η: The correlation ratio handles situations where one variable is categorical (e.g., groups/levels) and the relationship with the continuous variable may be nonlinear.
Why not Pearson/Spearman: Pearson targets linear association between continuous variables; Spearman targets monotonic rank association. Neither is purpose-built for “group vs. continuous” nonlinear structure like η.

In multiple regression, R=1 means the predicted values match the actual values exactly—zero residual error.

Explanation: Partial correlation quantifies the association between two variables while controlling for the effect of one or more additional variables.
Source: Live MCQ notes

Explanation: Adjusted R² penalizes the addition of unnecessary predictors, giving a more realistic measure of model performance. R² never decreases when you add predictors—even useless ones—so it can be inflated. Adjusted R² introduces a penalty based on sample size and number of predictors, rewarding variables that truly improve fit and discouraging pure noise.

Source: Applied General Statistics, Croxton and Cowden.

Pearson’s r only measures the strength and direction of linear relationships. If the relationship between two variables is non-linear, Pearson’s correlation might show r≈0 even though a strong non-linear association exists.
For example, a perfect parabolic relationship.

Spearman’s rank correlation uses ranks instead of raw data, making it less sensitive to outliers and applicable for ordinal data or non-linear monotonic relationships. Pearson’s correlation, in contrast, can be highly distorted by extreme values.
Source: Business Statistics,  SP Gupta, MP Gupta

Correlation does not imply causation. Here, both variables are likely related to hot weather — more ice creams are sold, and more people swim, increasing drowning risk. This is a classic example of spurious correlation caused by a third variable.

Explanation: r = 1.00 means a perfect positive linear relationship — all data points lie exactly on a straight line with a positive slope. It does not imply equal changes in absolute value (depends on slope), and it does not imply causation. If r = −1.00, the slope would be negative.

The regression equation is: Y = a + bX
We know that the regression line passes through the point (X̄, Ȳ)

​so, Ȳ = a + bX̄,
20 = a + (2)(10) 
​→ 20 = a + 20 
​→ a = 0.

Explanation:
Slope b = Σ(x−X̄)(y−Ȳ) / Σ(x−X̄)² = 48 / 32 = 1.5.
Intercept a = Ȳ − bX̄ = 12 − (1.5)(5) = 4.5.
Prediction: Ŷ = 4.5 + 1.5(8) = 16.5.

Explanation:
Formula: r__XY·Z = [r__XY − r__XZ·r__YZ] / sqrt[(1−r__XZ²)(1−r__YZ²)].
= [0.80 − (0.60)(0.50)] / sqrt[(1−0.36)(1−0.25)]
= 0.50 / sqrt(0.64·0.75) = 0.50 / 0.6928 ≈ 0.72.

In simple regression, R² = r²,
 so |r| = √0.64 = 0.80.
The sign of r matches the slope's sign; 
 since b < 0, r = − 0.80

Explanation:
In Y = a + bX, the slope b is the change in Y for a one-unit increase in X.
Here b = 0.8, so for 6 units increase:
Change in Y = 0.8 × 6 = 4.8 units.
Solution: Live MCQ class lecture

Explanation:
First, we have to find intercept a:
We know that, regression line passes through the (X̄,Ȳ)
we get, 100 = a + (1.2)(50) → a = 100 − 60 = 40.
Prediction: Ŷ = 40 + 1.2(60) = 40 + 72 = 112.

Multicollinearity occurs when two or more independent variables are highly correlated with each other, making it difficult to separate their individual effects on the dependent variable.
Source: Investopedia

Explanation:
R² = r², so |r| = √0.64 = 0.80.
Since b > 0, r = 0.80

Explanation:
The total variation in Y is measured by the Total Sum of Squares (SST)
SST = Σ(Y−Y¯)² = Total variation in Y
This total variation can be split into two parts:
SSR – Regression Sum of Squares: variation in Y explained by the regression on X
SSE – Error Sum of Squares: unexplained variation (residuals).

​Now, 
SSR = b² × Σ(x−X̄)² = (2.5)² × 40 = 6.25 × 40 = 250.
This is the explained variation in Y due to X.

Regression: Focuses on prediction and estimating the relationship between a dependent variable Y and one or more independent variables X
Correlation: correlation measures the strength and direction of association. Focuses on quantifying strength and direction of a linear relationship between two variables, without specifying which is dependent or independent.

​Source: Business Statistics, Md. Abdul Aziz, Regression Analysis: Montogomery.

The Correct statement could be-
"When measuring the degree of association between height and weight without predicting one from the other"
Explanation:

Why correlation here: If the goal is only to measure association strength without making predictions or designating dependent/independent variables, correlation is the right method.

Why not regression: Regression requires defining a dependent variable and is aimed at prediction or modeling relationships, not just describing association.

Option (c) describes multiple regression, not correlation.

Option (a) is regression territory, and (b) is also regression-focused.

Options (a) and (d) are wrong because the sign agreement between r and b holds in simple regression, but b can be any real number; it’s not restricted to [−1, +1].
Option (b) is wrong because neither method alone proves causation.

Regression coefficient b: Depends on the measurement units of X and Y (e.g., if X is in cm and Y in kg, b is kg/cm).

Correlation coefficient r: Standardized measure, unit-free, always between −1 and +1.

Classical regression assumptions include:

Linearity between X and Y

Homoscedasticity of residuals

Independence of residuals

Normality of residuals (for inference)
Perfect correlation (r=±1) is not an assumption; in fact, perfect correlation would be problematic because it would make predictions exact with no error (which rarely occurs in reality).

​Source: Regression Analysis, Montogomery.

Explanation:
The coefficient of determination R² = r²= (−0.90)²= 0.81
meaning 81% of the variation in exam mistakes can be explained by study time in the linear model.

​Option (a) is tempting, but correlation still does not prove causation.

​Option (b) is wrong because
r = −0.90 does not mean 90% variation explained—it’s 
R²

​Option (d) is wrong because “perfectly negative” means
r = −1, not −0.90

​Source:  Business Statistics, SP Gupta, MP Gupta, Live MCQ class lecture.

Spearman’s ρ is rank-based and captures monotonic (increasing or decreasing) trends regardless of their exact shape, making it more robust to nonlinearity.
Pearson’sr assumes linearity, so it may underestimate association if the relationship is nonlinear.

Correlation does not imply causation. A third variable—such as temperature—can drive both ice cream sales and swimming activity (which can lead to drownings).

This is a classic example of a lurking (confounding) variable causing a misleading correlation.

Options (a) and (b) confuse correlation with direct cause-effect; (d) is incorrect because there is a plausible explanation via a third factor.