Determine The Line Of Best Fit For The Data Shown Below. Round To The Nearest Hundredth.$\[ \begin{array}{|c|c|} \hline \text{Week} & \text{Number Of Complaints} \\ \hline 1 & 225 \\ 2 & 205 \\ 3 & 187 \\ 4 & 169 \\ 5 & 147
Introduction
In statistics, the line of best fit is a straight line that best represents the relationship between two variables in a dataset. It is also known as the regression line. The line of best fit is used to make predictions about the value of one variable based on the value of the other variable. In this article, we will determine the line of best fit for the given dataset.
Understanding the Dataset
The given dataset consists of the number of complaints for each week. The data is as follows:
| Week | Number of Complaints |
|---|---|
| 1 | 225 |
| 2 | 205 |
| 3 | 187 |
| 4 | 169 |
| 5 | 147 |
Calculating the Line of Best Fit
To calculate the line of best fit, we need to find the values of the slope (m) and the y-intercept (b). The equation of the line of best fit is given by:
y = mx + b
where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept.
Step 1: Calculate the Mean of x and y
To calculate the mean of x and y, we need to add up all the values of x and y and divide by the number of data points.
Mean of x = (1 + 2 + 3 + 4 + 5) / 5 = 3 Mean of y = (225 + 205 + 187 + 169 + 147) / 5 = 183
Step 2: Calculate the Slope (m)
The slope (m) is calculated using the formula:
m = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
where xi is each value of x, x̄ is the mean of x, yi is each value of y, and ȳ is the mean of y.
m = [(1-3)(225-183) + (2-3)(205-183) + (3-3)(187-183) + (4-3)(169-183) + (5-3)(147-183)] / [(1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)²] m = [(-2)(42) + (-1)(22) + (0)(4) + (1)(-14) + (2)(-36)] / [(-2)² + (-1)² + (0)² + (1)² + (2)²] m = [-84 - 22 + 0 - 14 - 72] / [4 + 1 + 0 + 1 + 4] m = -192 / 10 m = -19.2
Step 3: Calculate the y-Intercept (b)
The y-intercept (b) is calculated using the formula:
b = ȳ - m x̄
b = 183 - (-19.2) x 3 b = 183 + 57.6 b = 240.6
Equation of the Line of Best Fit
The equation of the line of best fit is given by:
y = mx + b y = -19.2x + 240.6
Rounding to the Nearest Hundredth
The line of best fit is rounded to the nearest hundredth as follows:
y = -19.20x + 240.60
Conclusion
Q: What is the line of best fit?
A: The line of best fit is a straight line that best represents the relationship between two variables in a dataset. It is also known as the regression line.
Q: Why is the line of best fit important?
A: The line of best fit is important because it can be used to make predictions about the value of one variable based on the value of the other variable. It can also be used to identify patterns and trends in the data.
Q: How is the line of best fit calculated?
A: The line of best fit is calculated using the following steps:
- Calculate the mean of x and y.
- Calculate the slope (m) using the formula: m = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
- Calculate the y-intercept (b) using the formula: b = ȳ - m x̄
- Write the equation of the line of best fit in the form: y = mx + b
Q: What is the difference between the line of best fit and the trend line?
A: The line of best fit and the trend line are both used to represent the relationship between two variables in a dataset. However, the line of best fit is a more precise representation of the relationship, while the trend line is a more general representation.
Q: Can the line of best fit be used to make predictions about future data?
A: Yes, the line of best fit can be used to make predictions about future data. However, it is essential to note that the accuracy of the predictions will depend on the quality of the data and the complexity of the relationship between the variables.
Q: How can the line of best fit be used in real-world applications?
A: The line of best fit can be used in a variety of real-world applications, including:
- Predicting sales or revenue based on marketing efforts
- Identifying trends in stock prices or other financial data
- Analyzing the relationship between variables in a scientific experiment
- Making predictions about the performance of a product or service
Q: What are some common mistakes to avoid when determining the line of best fit?
A: Some common mistakes to avoid when determining the line of best fit include:
- Not checking for outliers or anomalies in the data
- Not using a sufficient number of data points
- Not considering the complexity of the relationship between the variables
- Not using a robust method for calculating the line of best fit
Q: How can the line of best fit be visualized?
A: The line of best fit can be visualized using a variety of methods, including:
- Plotting the data points and the line of best fit on a graph
- Using a scatter plot to show the relationship between the variables
- Creating a residual plot to show the difference between the observed and predicted values
Q: What are some common tools and software used to determine the line of best fit?
A: Some common tools and software used to determine the line of best fit include:
- Microsoft Excel
- Google Sheets
- R
- Python
- SPSS
Conclusion
In this article, we have answered some frequently asked questions about determining the line of best fit. We have discussed the importance of the line of best fit, how it is calculated, and how it can be used in real-world applications. We have also highlighted some common mistakes to avoid and provided information on how to visualize the line of best fit.