Edexcel IAL S1 October 2023 question paper and mark scheme (WST01/01)

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1. Data and Summary Statistics · 9 marks

   The mean of three numbers, \(a\), \(b\), and \(c\), is exactly \(8\). The variance of these three numbers is \(10\).

   1. Using the formula for variance, prove that \(a^2 + b^2 + c^2 = 222\).

   2. Given that \(c = 10\), find the mean and the variance of the remaining two numbers, \(a\) and \(b\).

   Worked answer

   1. Let the sum of squares be \(\sum x^2 = a^2 + b^2 + c^2\).
      Using the variance formula: \(\text{Var}(X) = \frac{\sum x^2}{n} - \mu^2\). M1
      \(10 = \frac{a^2 + b^2 + c^2}{3} - 8^2\). M1
      \(10 = \frac{a^2 + b^2 + c^2}{3} - 64 \implies 74 = \frac{a^2 + b^2 + c^2}{3}\). A1
      \(a^2 + b^2 + c^2 = 3 \times 74 = 222\). A1 *

   2. If \(c = 10\), then \(a + b + 10 = 3 \times 8 = 24 \implies a + b = 14\). M1
      New mean for just \(a\) and \(b\): \(\frac{14}{2} = 7\). A1
      Substitute \(c=10\) into sum of squares: \(a^2 + b^2 + 100 = 222 \implies a^2 + b^2 = 122\). M1
      New Variance for \(a\) and \(b\) \(= \frac{a^2 + b^2}{2} - (\text{New Mean})^2\). M1
      \(= \frac{122}{2} - 7^2 = 61 - 49 = 12\). A1

2. Correlation and Regression · 4 marks

   Using the summary statistics $\sum x = 140$, $\sum y = 200$, $S_{xx} = 270$, $S_{xy} = -300$, and $n=20$, find the equation of the regression line of $y$ on $x$ in the form $y = a + bx$.

   Worked answer

   1. M1 for calculating the gradient $b = \frac{S_{xy}}{S_{xx}} = \frac{-300}{270}$.
   2. A1 for $b = -\frac{10}{9}$ or $\approx -1.11$.
   3. M1 for calculating the means $\bar{x} = 7$, $\bar{y} = 10$, and determining the intercept $a = 10 - (-\frac{10}{9})(7)$.
   4. A1 for final equation $y = 17.8 - 1.11x$ (or $y = \frac{160}{9} - \frac{10}{9}x$).

3. Measures of Location and Spread · 5 marks

   For a sample of $n=10$ observations, $\Sigma x = 120$ and $\Sigma x^2 = 1800$. A new variable $y$ is defined as $y = 3x - 2$. Calculate the mean and variance of $y$.

   Worked answer

   1. M1 for calculating $\bar{x} = \frac{120}{10} = 12$.
   2. M1 for finding $\bar{y} = 3(12) - 2 = 34$.
   3. M1 for calculating $Var(x) = \frac{1800}{10} - 12^2 = 180 - 144 = 36$.
   4. M1 for applying the variance transformation rule $Var(y) = 3^2 \times Var(x)$.
   5. A1 for finding $Var(y) = 9 \times 36 = 324$.

More on these topics: Data and Summary Statistics · Correlation and Regression · Measures of Location and Spread

S1 topics this paper can test

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• Data and Summary Statistics
• Representation of Data
• Probability
• Correlation and Regression
• Discrete Random Variables
• Normal Distribution
• Measures of Location and Spread

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Formula booklet

Mathematical Formulae and Statistical Tables (International AS/A Level Mathematics) · Pearson Edexcel · For the 2018 IAL specification (current)