The problem provides a table showing the performance of 10 students in Chemistry ($x$) and Physics ($y$). The task involves: (a) Plotting a scatter diagram. (b) Calculating the means of $x$ and $y$, denoted as $\bar{x}$ and $\bar{y}$ respectively. Also, calculating the means of $x$ and $y$ values above $\bar{x}$, denoted as $(x_1, y_1)$. (c) Drawing the line of best fit through $(\bar{x}, \bar{y})$ and $(x_1, y_1)$. (d) Using the graph, determining the relationship between $x$ and $y$, and finding the value of $y$ when $x = 77$. I will solve part (b) only.

1. Problem Description

The problem provides a table showing the performance of 10 students in Chemistry ($x$) and Physics ($y$). The task involves:

(a) Plotting a scatter diagram.

(b) Calculating the means of $x$ and $y$, denoted as $\bar{x}$ and $\bar{y}$ respectively. Also, calculating the means of $x$ and $y$ values above $\bar{x}$, denoted as $(x_1, y_1)$.

(c) Drawing the line of best fit through $(\bar{x}, \bar{y})$ and $(x_1, y_1)$.

(d) Using the graph, determining the relationship between $x$ and $y$, and finding the value of $y$ when $x = 77$.

I will solve part (b) only.

2. Solution Steps

(b) (i) Calculating the means of $x$ and $y$:

The formula for the mean is:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

$\bar{y} = \frac{\sum_{i=1}^{n} y_i}{n}$

Here, $n = 10$.

The values of $x$ are: 30, 55, 60, 65, 70, 74, 75, 84, 87,

9

0. The values of $y$ are: 55, 65, 75, 79, 83, 73, 69, 85, 90,

8

6.

$\sum_{i=1}^{10} x_i = 30 + 55 + 60 + 65 + 70 + 74 + 75 + 84 + 87 + 90 = 690$

$\sum_{i=1}^{10} y_i = 55 + 65 + 75 + 79 + 83 + 73 + 69 + 85 + 90 + 86 = 760$

$\bar{x} = \frac{690}{10} = 69$

$\bar{y} = \frac{760}{10} = 76$

(b) (ii) Calculating the means of $x$ and $y$ values above $\bar{x}$:

$\bar{x} = 69$. We need to find the $x$ and $y$ values where $x > 69$.

The $x$ values greater than 69 are: 70, 74, 75, 84, 87,

9

0. The corresponding $y$ values are: 83, 73, 69, 85, 90,

8

6.

There are 6 values.

$\sum x_i = 70 + 74 + 75 + 84 + 87 + 90 = 480$

$\sum y_i = 83 + 73 + 69 + 85 + 90 + 86 = 486$

$x_1 = \frac{480}{6} = 80$

$y_1 = \frac{486}{6} = 81$

3. Final Answer

(b) (i) $\bar{x} = 69$, $\bar{y} = 76$

(b) (ii) $(x_1, y_1) = (80, 81)$