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.