Let's break down and solve the regression problem provided.

The problem involves fitting a linear regression model of the form $y = a + bx$, where:

• $y$ is the total cost (in £'000s),
• $x$ is the output (in thousands of boxes).

We are given the following dataLet's break down and solve the regression problem provided.

The problem involves fitting a linear regression model of the form $y = a + bx$, where:

• $y$ is the total cost (in £'000s),
• $x$ is the output (in thousands of boxes).

We are given the following

Steps to Solve:

a) Calculate the regression equation $y = a + bx$

1. Find the necessary summations:

   • $\sum x$ (sum of all $x$),
   • $\sum y$ (sum of all $y$),
   • $\sum x^2$ (sum of squares of $x$),
   • $\sum xy$ (sum of the product of $x$ and $y$).

2. Calculate the slope $b$ using the formula: $b = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2}$ where $n$ is the number of data points (in this case, $n = 10$).

3. Calculate the intercept $a$ using the formula: $a = \frac{\sum y - b \sum x}{n}$

b) Interpret the regression coefficients $a$ and $b$

• $a$ is the intercept, representing the estimated total cost when no output (boxes produced) is made.
• $b$ is the slope, representing the rate of change of the total cost for each additional thousand boxes produced.

c) Calculate and interpret the correlation coefficient $r$

The correlation coefficient $r$ measures the strength and direction of the linear relationship between $x$ (output) and $y$ (total cost). It can be calculated using: [ r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt