Regression Analysis: Simple Regression, Equations and Uses Questions

Question 1

Define regression analysis.

Accepted Answer

Regression analysis is a statistical method that estimates the relationship between variables and helps predict a dependent variable from an independent variable.

Question 2

Write the simple regression equation of Y on X.

Accepted Answer

A simple regression equation is Y = a + bX.

Question 3

In Y = a + bX, what is “a”?

Accepted Answer

a is the intercept: the value of Y when X = 0.

Question 4

In Y = a + bX, what is “b”?

Accepted Answer

b is the slope/regression coefficient showing change in Y for a unit change in X.

Question 5

What is least squares method?

Accepted Answer

Least squares is a method of fitting a regression line by minimizing the sum of squared deviations of observed values from fitted values.

Question 6

Regression is mainly used for what purpose?

Accepted Answer

Regression is mainly used for prediction/forecasting.

Question 7

Differentiate between correlation and regression (any three points).

Accepted Answer

Correlation and regression differ as follows:

| Point | Correlation | Regression |
|---|---|---|
| Purpose | measures association | provides prediction equation |
| Output | r (−1 to +1) | regression line/equation |
| Variables | treated equally | dependent and independent identified |

(Any three points can be written.)

Question 8

Write any three uses of regression in business.

Accepted Answer

Uses of regression in business include:

1) Predicting sales from advertising expenditure.
2) Predicting demand from price/income.
3) Estimating cost from output level (cost estimation).

(Any three uses can be written.)

Question 9

Explain regression coefficient (any three points).

Accepted Answer

Regression coefficient points:

1) Regression coefficients (b) measure the rate of change between variables.
2) b_yx is used to predict Y from X; b_xy predicts X from Y.
3) They are affected by the units of measurement (change of scale changes b).

(Any three points can be written.)

Question 10

Explain least squares method (any three points).

Accepted Answer

Least squares method:

1) Fits the “best” line through data points.
2) Minimizes the sum of squared residuals (observed Y − predicted Y)².
3) Provides a, b in Y = a + bX for prediction.

(Any three points can be written.)

Question 11

Write any three precautions/limitations of regression.

Accepted Answer

Precautions/limitations include:

1) Do not assume causation; regression shows association/prediction.
2) Avoid extrapolation beyond observed data range.
3) Outliers and unstable relationships can distort the regression line.

(Any three points can be written.)

Question 12

State the relationship between r and regression coefficients.

Accepted Answer

Relationship: r = ±√(b_xy × b_yx). Hence r² = b_xy × b_yx.

Question 13

Explain simple linear regression and interpret the regression equation with a worked example.

Accepted Answer

Simple linear regression studies the relationship between two variables and provides a prediction equation. When Y depends on X, the regression equation is written as:

Y = a + bX

where a is the intercept and b is the slope (regression coefficient). The slope tells how much Y is expected to change when X increases by one unit.

**Worked example:**
Suppose the regression equation is:

Y = 20 + 3X

If X = 10, then predicted Y = 20 + 3(10) = 50.

Interpretation:
- Intercept 20: the base value of Y when X is zero.
- Slope 3: for each 1 unit rise in X, Y increases by 3 units.

Thus, regression is very useful for business forecasting, such as predicting sales (Y) from advertising (X).

Question 14

Differentiate correlation and regression using a table and explain why regression is used for prediction.

Accepted Answer

Correlation and regression both study relationship between variables, but their purpose and output are different.

| Point | Correlation | Regression |
|---|---|---|
| Main aim | measure degree of association | estimate equation for prediction |
| Output | r (−1 to +1) | Y = a + bX (regression line) |
| Variable roles | no dependent/independent | dependent (Y) and independent (X) |
| Unit effect | r is unit-free | b depends on units |

Correlation only tells whether the variables move together and how strongly. It does not tell how much Y will change for a given change in X.

Regression provides a specific equation. Using the equation, we can substitute X and predict Y. This is why regression is widely used in business for sales forecasting, cost estimation and demand prediction.

Therefore, regression is the correct tool when the objective is prediction.

Question 15

Explain least squares method and steps to fit a regression line. Use a flowchart.

Accepted Answer

Least squares is the most common method for fitting a regression line. It chooses the line that best fits the data by minimizing the sum of squared differences between the observed values and the values predicted by the line.

```mermaid
flowchart LR
  A[Collect paired data (X,Y)] --> B[Compute X̄ and Ȳ]
  B --> C[Compute required sums
(ΣX, ΣY, ΣXY, ΣX²)]
  C --> D[Find slope b and intercept a]
  D --> E[Write regression equation
Y = a + bX]
  E --> F[Predict Y for given X]
```

The key idea is: the “best” line is the one with minimum Σ(Observed Y − Predicted Y)².

Least squares regression is widely used because it is objective, uses all observations and gives the most reliable straight-line fit for prediction when the relationship is approximately linear.

Thus, least squares provides a practical way to obtain regression equation for business forecasting and decision making.

Point	Correlation	Regression
Purpose	measures association	provides prediction equation
Output	r (−1 to +1)	regression line/equation
Variables	treated equally	dependent and independent identified

Regression Analysis: Simple Regression, Equations and Uses

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