Edexcel IAL S1 October 2020 question paper and mark scheme (WST01/01)

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1. Data and Summary Statistics · 9 marks

   Two different groups of students took a memory test. Group A consists of \(20\) students. Their mean score is \(15\) and the standard deviation is \(3\). Group B consists of \(30\) students. Their mean score is \(20\) and the standard deviation is \(4\). The two groups are combined into a single group of \(50\) students.

   1. Find the combined mean score.

   2. By calculating the total sum of squares (\(\sum x^2\)) for each group separately, find the overall standard deviation of the combined group.

   Worked answer

   1. Sum A \(= 20 \times 15 = 300\). Sum B \(= 30 \times 20 = 600\). M1
      Total Sum \(= 300 + 600 = 900\). Total students \(= 50\).
      Combined mean \(= \frac{900}{50} = 18\). A1

   2. For Group A: \(\sigma_A^2 = 3^2 = 9\).
      \(9 = \frac{\sum x_A^2}{20} - 15^2 \implies \frac{\sum x_A^2}{20} = 9 + 225 = 234 \implies \sum x_A^2 = 4680\). M1A1
      For Group B: \(\sigma_B^2 = 4^2 = 16\).
      \(16 = \frac{\sum x_B^2}{30} - 20^2 \implies \frac{\sum x_B^2}{30} = 16 + 400 = 416 \implies \sum x_B^2 = 12480\). M1A1
      Total sum of squares \(\sum x^2 = 4680 + 12480 = 17160\). A1
      Combined Variance \(= \frac{17160}{50} - 18^2 = 343.2 - 324 = 19.2\). M1
      Combined SD \(= \sqrt{19.2} = 4.3817... \implies 4.38\) (3sf). A1

2. Correlation and Regression · 4 marks

   A dataset of 20 observations produced the following summary statistics: $\sum x = 140$, $\sum y = 200$, $\sum x^2 = 1250$, $\sum y^2 = 2600$, and $\sum xy = 1100$. Calculate the product moment correlation coefficient.

   Worked answer

   1. M1 for calculating $S_{xx} = 1250 - \frac{140^2}{20} = 270$ and $S_{yy} = 2600 - \frac{200^2}{20} = 600$.
   2. M1 for calculating $S_{xy} = 1100 - \frac{140 \times 200}{20} = -300$.
   3. M1 for setting up the PMCC calculation $r = \frac{-300}{\sqrt{270 \times 600}}$.
   4. A1 for evaluating the final correlation coefficient as $r \approx -0.745$ (to 3 s.f.).

3. Measures of Location and Spread · 4 marks

   A dataset $x$ has mean $\bar{x} = 5$ and standard deviation $\sigma_x = 2$. A new dataset $y$ is formed using the linear transformation $y = ax + b$, where $a > 0$. The new dataset has mean $\bar{y} = 15$ and standard deviation $\sigma_y = 6$. Find the values of $a$ and $b$.

   Worked answer

   1. M1 for linking the standard deviations: $\sigma_y = a \sigma_x$.
   2. A1 for solving for $a$: $6 = a(2) \implies a = 3$.
   3. M1 for linking the means: $\bar{y} = a\bar{x} + b$.
   4. A1 for substituting $a=3$ to find $b$: $15 = 3(5) + b \implies b = 0$.

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

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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)