Class 11 Economics (Statistics for Economics) Chapter 6 Correlation

Key Concept: Pearson’s Coefficient Range

b) -1 to 1

[Solution Description] The range of Karl Pearson’s coefficient of correlation is from -1 to 1. This indicates the strength and direction of a linear relationship between two variables, where -1 denotes a perfect negative linear relationship, 0 means no linear relationship, and 1 denotes a perfect positive linear relationship.

Key Concept: Basic Example Identification

b) Income and consumption

[Solution Description] A positive correlation is present when an increase in one variable results in an increase in another. Income and consumption often increase together, exhibiting a positive correlation.

Key Concept: Basic Example Analysis

a) Positive correlation

[Solution Description] Generally, more hours studied correlate positively with higher exam scores, showing a positive correlation.

Key Concept: Change of Origin and Scale

b) Origin and scale

[Solution Description] The magnitude of the correlation coefficient $r$ is unaffected by changes in origin and scale, which means that adding or multiplying each data point by a constant does not change $r$.

Key Concept: Example Identification

c) As study time increases, failure chances decrease

[Solution Description] A negative correlation is evident when an increase in one variable results in a decrease in the other. For instance, the more time spent studying (study time increases), the fewer chances there are of failing (failure chances decrease).

Key Concept: Correlation and Causation

c) Correlation measures covariation

[Solution Description] Correlation measures the degree to which two variables move in relation to each other (covariation), but it does not imply that changes in one cause changes in the other (causation).

Key Concept: Basic Definition

b) They have no linear relationship

[Solution Description] A zero correlation coefficient indicates that there is no linear relationship between the two variables. This means changes in one variable do not predict or infer changes in the other.

Key Concept: Definition Recall

b) Negative Correlation

[Solution Description] When variables move in opposite directions, it indicates a negative correlation. For instance, if one variable increases while another decreases, this shows an inverse relationship.

Key Concept: Properties of Coefficient

d) It ranges between -1 and 1 inclusive.

[Solution Description] Spearman’s rank correlation coefficient has several properties: it lies between -1 and 1, it is a non-parametric measure, and it is not affected by extreme values like Pearson’s correlation might be. Considering these properties helps identify the correct statement.

Key Concept: Application of Formula

d) -1

[Solution Description] To find the Spearman’s rank correlation coefficient $r_s$, we use the formula:

$$r_s = 1 – \frac{6\sum D^2}{n(n^2-1)}$$

where $D$ is the difference between corresponding ranks, and $n$ is the number of observations.

1. Calculate $D$:

$D_1 = 3 – 2 = 1$

$D_2 = 1 – 4 = -3$

$D_3 = 4 – 1 = 3$

$D_4 = 2 – 3 = -1$

2. Square the differences and sum them:

$D^2_1 = 1^2 = 1$

$D^2_2 = (-3)^2 = 9$

$D^2_3 = 3^2 = 9$

$D^2_4 = (-1)^2 = 1$

$D^2 = 1 + 9 + 9 + 1 = 20$

3. Substitute into the formula:

$$r_s = 1 – \frac{6 \times 20}{4(4^2-1)} = 1 – \frac{120}{60} = 1 – 2 = -1$$

Therefore, the Spearman’s rank correlation coefficient is -1.

Key Concept: Direction of Relationship

b) In opposite directions

[Solution Description] A negative correlation indicates that as one variable increases, the other decreases. So, the variables move in opposite directions when there is a negative correlation.

Key Concept: Rank Calculation

c) 0.95

[Solution Description] To calculate the Spearman’s rank correlation coefficient, assign ranks to each variable first. For X: – X = 10 corresponds to Rank 1 – X = 15 corresponds to Rank 2.5 (average of ranks 2 and 3) – X = 20 corresponds to Rank 4 For Y: – Y = 1 corresponds to Rank 1 – Y = 2 corresponds to Rank 2 – Y = 3 corresponds to Rank 3 – Y = 4 corresponds to Rank 4 Calculate $D^2$ for each pair: – D (X ranked 1, Y ranked 1) = 0, thus $D^2 = 0$ – D (X ranked 2.5, Y ranked 2) = 0.5, thus $D^2 = 0.25$ – D (X ranked 2.5, Y ranked 3) = 0.5, thus $D^2 = 0.25$ – D (X ranked 4, Y ranked 4) = 0, thus $D^2 = 0$ $\Sigma D^2 = 0 + 0.25 + 0.25 + 0 = 0.5$ $n = 4$, apply formula: $r = 1 – \frac{6 \times 0.5}{4(4^2-1)} = 1 – \frac{3}{60} = 1 – 0.05 = 0.95$ Thus, the rank correlation coefficient is 0.95.

Key Concept: Scatter Diagram Basics

c) No correlation

[Solution Description] When a scatter diagram shows points that are widely scattered with no apparent pattern, it indicates no correlation between the variables. This means changes in one variable do not predictably affect the other.

Key Concept: Positive/Negative Correlation

b) As the amount of exercise increases, weight decreases.

[Solution Description] A negative correlation occurs when one variable increases, and the other decreases. For example, as the price of a product increases, its demand tends to decrease.

Key Concept: Correlation Interpretation

c) Strong negative linear relation

[Solution Description] The correlation coefficient value can range from -1 to +1. A value of -0.8 indicates a strong negative linear relationship between the variables. This means as one variable increases, the other tends to decrease significantly.

Key Concept: Simple Example

d) Temperature and ice cream sales

[Solution Description] A positive correlation example involves situations where an increase in one variable tends to be associated with an increase in another variable. An example is higher temperature leading to increased ice cream sales.

Key Concept: Correlation Coefficient

d) Strong negative correlation

[Solution Description] The correlation coefficient value of -0.8 indicates a strong negative correlation, meaning as one variable increases, the other tends to decrease significantly.

Key Concept: Scatter Diagram Interpretation

c) No correlation

[Solution Description] A scatter diagram with widely scattered points without a clear pattern suggests no correlation between the variables. In such cases, there is neither a positive nor negative relationship apparent.

Key Concept: Positive vs Negative

b) Negative correlation

[Solution Description] This scenario is an example of negative correlation. In negative correlation, one variable increases while the other decreases, indicating an inverse relationship.

Key Concept: Scatter Diagram Interpretation

c) No correlation

[Solution Description] When the points on a scatter diagram are spread randomly with no clear pattern, it indicates no correlation between the variables.

Key Concept: Correlation Definition

c) One variable increases while the other decreases

[Solution Description] Negative correlation occurs when one variable increases while the other decreases. It represents an inverse relationship between two variables.

Key Concept: Rank Correlation Preference

b) When variables cannot be measured precisely

[Solution Description] Spearman’s rank correlation is used when the variables cannot be measured precisely or involve ordinal data, which does not require normal distribution assumptions like Pearson’s correlation does.

Key Concept: Basic Concept

d) Variables move in the same direction

[Solution Description] A positive value of Karl Pearson’s Coefficient of Correlation suggests that the two variables in question tend to move in the same direction. This means that as one variable increases, the other variable also tends to increase.

Key Concept: Correlation Coefficient Interpretation

c) Strong negative correlation

[Solution Description] The correlation coefficient $r$ ranges from -1 to +1. A value close to -1 implies a strong negative correlation, meaning that as one variable increases, the other tends to decrease strongly.

Key Concept: Correlation Types

d) Increased exercise leads to weight loss.

[Solution Description] A negative correlation occurs when one variable increases while the other decreases. In this scenario, as you increase your level of physical activity, your weight tends to decrease.

Key Concept: Concept Check

c) The correlation coefficient ranges between -1 and 1.

[Solution Description] The correlation coefficient ranges between -1 and 1, indicating the strength and direction of the linear relationship between two variables.

Key Concept: Definition Recall

b) -1 to 1

[Solution Description] The Pearson correlation coefficient $r$ can take any value from -1 to 1, inclusive. This indicates the direction and strength of a linear relationship between two variables.