Mathematics Examination Structure

Examination Structure

Duration: 2 hours 40 minutes, including 10 minutes reading time
Total marks: 75
Section A: Answer all questions (35 marks total)
Section B: Answer 2 questions (40 marks total)
Use of an approved calculator permitted
Show all workings; examination materials cannot be removed

Section A Overview

Question A1:
- Calculate new average price after a 15% markdown on £200 products.
- Calculate new average price after adding items priced at £135, £162, £148, and £187 with markdowns.
Question A2:
- Define market equilibrium for supply and demand.
- Given supply function (Q_s = 3P - 120) and demand function (Q_D = 600 - 2P), find equilibrium price and quantity.
Question A3:
- Pearson correlation coefficient of 0.60 suggests a moderate positive correlation.
- Sketch a scatter diagram and regression line for variables (Y) on (X) indicating this correlation.
Question A4:
- Profit function given as (P(q) = 80q + 150q^2 - 14q^3).
- Determine (q) for maximum profit and the threshold of profitability.
Question A5:
- Average costs for goods in 2021: £159, £76, £59.
- Costs for 2022: £162, £69, £64; relative amounts sold: 5, 24, 10.
- Calculate and interpret the Paasche index for these goods.
Question A6:
- Marginal cost function (MC(Q) = 800 - 4Q).
- Determine fixed cost given a maximum total cost of £100,000.

Section B Overview

Question B1:
- Analyze marks of 5000 students using frequency distribution.
- Calculate mean and standard deviation, discuss results.
- Use Pearson’s Skewness Coefficient to assess skewness.
- Evaluate whether 80% of students scored between 35 and 75.
Question B2:
- Given covariances and means, calculate the regression line equation for (Y) on (X).
- Determine and interpret the correlation coefficient between (Y) and (X); assess prediction reliability.
- Compare scatter diagrams for correlation strength and identify relationship types.
Question B3:
- Analyze elasticity of demand with the given function.
- Derive demand function for good (x) with specified (E(p)) and revenue function based on quantity demanded (q).
- Identify maximum revenue point and corresponding quantity (q).

Key Concepts to Focus On

Market equilibrium: Point where supply equals demand.
Correlation vs. causation: Pearson's coefficient indicates relationship strength.
Profit functions: Analyzing derivatives to find maxima and break-even points.
Index calculations: Understanding price changes over time through indices like the Paasche index.
Elasticity and revenue: Demand elasticity influences pricing and revenue strategies.

Examination Structure

Duration: 2 hours 40 minutes, including 10 minutes reading time
Total marks: 75
Section A: Answer all questions (35 marks total)
Section B: Answer 2 questions (40 marks total)
Use of an approved calculator permitted
Show all workings; examination materials cannot be removed

Section A Overview

Question A1:
- Calculate new average price after a 15% markdown on £200 products.
- Calculate new average price after adding items priced at £135, £162, £148, and £187 with markdowns.
Question A2:
- Define market equilibrium for supply and demand.
- Given supply function (Q_s = 3P - 120) and demand function (Q_D = 600 - 2P), find equilibrium price and quantity.
Question A3:
- Pearson correlation coefficient of 0.60 suggests a moderate positive correlation.
- Sketch a scatter diagram and regression line for variables (Y) on (X) indicating this correlation.
Question A4:
- Profit function given as (P(q) = 80q + 150q^2 - 14q^3).
- Determine (q) for maximum profit and the threshold of profitability.
Question A5:
- Average costs for goods in 2021: £159, £76, £59.
- Costs for 2022: £162, £69, £64; relative amounts sold: 5, 24, 10.
- Calculate and interpret the Paasche index for these goods.
Question A6:
- Marginal cost function (MC(Q) = 800 - 4Q).
- Determine fixed cost given a maximum total cost of £100,000.

Section B Overview

Question B1:
- Analyze marks of 5000 students using frequency distribution.
- Calculate mean and standard deviation, discuss results.
- Use Pearson’s Skewness Coefficient to assess skewness.
- Evaluate whether 80% of students scored between 35 and 75.
Question B2:
- Given covariances and means, calculate the regression line equation for (Y) on (X).
- Determine and interpret the correlation coefficient between (Y) and (X); assess prediction reliability.
- Compare scatter diagrams for correlation strength and identify relationship types.
Question B3:
- Analyze elasticity of demand with the given function.
- Derive demand function for good (x) with specified (E(p)) and revenue function based on quantity demanded (q).
- Identify maximum revenue point and corresponding quantity (q).

Key Concepts to Focus On

Market equilibrium: Point where supply equals demand.
Correlation vs. causation: Pearson's coefficient indicates relationship strength.
Profit functions: Analyzing derivatives to find maxima and break-even points.
Index calculations: Understanding price changes over time through indices like the Paasche index.
Elasticity and revenue: Demand elasticity influences pricing and revenue strategies.

Correlation Coefficient

Measures the strength and direction of a linear relationship between two variables.
Pearson's correlation coefficient (r) is calculated using the formula:
- ( r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} )
Correlation values range from -1 to +1:
- ( r = 1 ): Perfect positive correlation
- ( r = -1 ): Perfect negative correlation
- ( r = 0 ): No correlation
Strong correlation is indicated by values close to 1 or -1, while values near 0 suggest weakness.
Positive ( r ) signifies that as one variable increases, the other also tends to increase, and vice versa for negative ( r ).

Simple Linear Regression

A method for modeling the relationship between two variables using a linear equation.
The regression equation is defined as:
- ( Y = b_0 + b_1X + \epsilon )
  - ( Y ): Dependent variable
  - ( X ): Independent variable
  - ( b_0 ): Y-intercept (value of ( Y ) when ( X = 0 ))
  - ( b_1 ): Slope of the regression line (change in ( Y ) for a unit change in ( X ))
  - ( \epsilon ): Error term
Coefficients are calculated as follows:
- Slope (( b_1 )):
  - ( b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2} )
- Intercept (( b_0 )):
  - ( b_0 = \bar{y} - b_1\bar{x} )
Key assumptions include linearity, independence of residuals, homoscedasticity (constant variance of errors), and normality of errors.

Interpreting Regression Outputs

Coefficients provide insights into the relationship:
- ( b_0 ): Predicted value of ( Y ) when ( X ) is zero.
- ( b_1 ): Represents the expected change in ( Y ) corresponding to a one-unit increase in ( X ).
R-squared (R²) quantifies the proportion of variance in the dependent variable accounted for by the independent variable:
- Values range from 0 (no fit) to 1 (perfect fit).
P-values assess the statistical significance of the coefficients:
- A p-value < 0.05 indicates that the coefficient is significantly different from zero.
Confidence Intervals provide a range for the true parameter value:
- A 95% confidence interval is commonly used; if it does not include zero, the coefficient is significant.

Correlation Coefficient

Measures the strength and direction of a linear relationship between two variables.
Types:
- Pearson's r: Ranges from -1 to 1 indicating:
  - 1 signifies perfect positive correlation
  - -1 signifies perfect negative correlation
  - 0 signifies no correlation
- Spearman's rank correlation: Non-parametric measure for monotonic relationships.
Pearson’s r Formula:
( r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} )
Characteristics:
- Sensitive to outliers which can distort results.
- Assumes linearity, normal distribution, and constant variance (homoscedasticity).

Simple Linear Regression

Models the relationship between two variables using a linear equation.
Regression Equation:
( Y = b_0 + b_1X )
- ( Y ) denotes the dependent variable (response).
- ( X ) denotes the independent variable (predictor).
- ( b_0 ) is the Y-intercept.
- ( b_1 ) signifies the slope of the line.
Execution Steps:
- Calculate slope ( b_1 ) and intercept ( b_0 ).
- Utilize the least squares method to minimize the sum of squared residuals.
Assumptions:
- Linearity: Assumes a linear relationship between X and Y.
- Independence: Assumes independent observations.
- Homoscedasticity: Assumes constant variance of residuals.
- Normality: Assumes the residuals are normally distributed.

Interpreting Regression Outputs

Coefficients:
- ( b_0 ): Estimated value of Y when X equals zero.
- ( b_1 ): Change in Y correlated with a one-unit change in X.
R-squared (R²):
- Indicates the proportion of variance in the dependent variable explained by the independent variable.
- Values range from 0 to 1; higher values indicate a better model fit.
P-values:
- Used to assess the significance of regression coefficients.
- A p-value less than 0.05 typically indicates statistical significance.
Residual Analysis:
- Essential for examining residuals for patterns; should appear randomly dispersed.
- Outliers can signify data points that may not conform to the model well.
Confidence Intervals:
- Provide a range of values for regression coefficients, reflecting reliability and precision of estimates.