https://file.aiquickdraw.com/mathbot/file/1728397689380zzcu8scy.jpg

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 

| Week | Output \( x \) | Total cost \( y \) |
|------|----------------|---------------------|
| 1    | 20             | 60                  |
| 2    | 2              | 25                  |
| 3    | 3              | 26                  |
| 4    | 23             | 66                  |
| 5    | 18             | 49                  |
| 6    | 14             | 48                  |
| 7    | 10             | 35                  |
| 8    | 8              | 18                  |
| 9    | 13             | 40                  |
| 10   | 8              | 33                  |

### 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

A cost accountant has derived the following data on the weekly output of standard size boxes from a factory.
Week | Output x (thousands of boxes) | Total cost y (£'000s)
1 | 20 | 60
2 | 2 | 25
3 | 3 | 26
4 | 23 | 66
5 | 18 | 49
6 | 14 | 48
7 | 10 | 35
8 | 8 | 18
9 | 13 | 40
10 | 8 | 33

Required:

a) Calculate the regression equation y = a + bx.
b) Interpret the regression coefficients a and b.
c) Calculate and interpret the correlation coefficient r.

Learn how to calculate the linear regression equation y = a + bx using factory box output data. This problem involves interpreting regression coefficients and finding the correlation coefficient, making it ideal for understanding key concepts in statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation