how to calculate residuals on ti 84

Everly

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19-01-2025

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Calculating residuals is a crucial step in regression analysis. Residuals represent the difference between the observed values and the values predicted by a regression model. Understanding residuals helps assess the accuracy and goodness of fit of your model. This guide shows you how to easily calculate residuals using your TI-84 calculator.

Understanding Residuals

Before diving into the calculations, let's clarify what residuals are. In a regression model (like linear regression), you're trying to find the best-fitting line or curve that describes the relationship between your variables. The equation for a linear regression is typically represented as: ŷ = mx + b, where ŷ is the predicted value, m is the slope, x is the independent variable, and b is the y-intercept.

A residual is the difference between the actual observed value (y) and the predicted value (ŷ): Residual = y - ŷ. Positive residuals indicate that the observed value is above the predicted value, while negative residuals mean the observed value is below the predicted value.

Calculating Residuals on your TI-84

Here's a step-by-step guide on how to calculate residuals using your TI-84 calculator after performing a linear regression:

1. Enter Your Data:

• Press STAT then select 1:Edit.
• Enter your independent variable data (x-values) into L1 and your dependent variable data (y-values) into L2.

2. Perform Linear Regression:

• Press STAT, then move to the CALC menu.
• Select 4:LinReg(ax+b).
• Make sure that Xlist is set to L1 and Ylist is set to L2. You can adjust these if your data is in different lists.
• Press ENTER. The calculator will display the equation of the regression line (a = slope, b = y-intercept), along with other statistical information like R² (coefficient of determination).

3. Access Residuals:

• This is where it gets slightly different depending on your TI-84 model and operating system. Here's the most common approach:

  • Press 2nd then STAT PLOT (Y=).
  • Select Plot1 (or whichever plot you want to use).
  • Turn the plot ON.
  • Choose the scatter plot type.
  • Set Xlist to L1 and Ylist to RESID. You'll find RESID by pressing 2nd then STAT, then scrolling to the RESID option.

4. View the Residuals:

• Press GRAPH. The calculator will display the scatter plot of your data, with the points visually representing their residuals based on the regression line.

5. Access Residual Values Numerically:

• To see the actual numerical values of the residuals:

  • Press STAT then 1:Edit.
  • You should now see a list labeled RESID (or similar depending on your OS version). This list contains the calculated residuals for each data point.

Important Note: Some newer TI-84 models might have a slightly different menu structure or require additional steps. If you encounter difficulties, consult your calculator's manual or search online for tutorials specific to your TI-84 model and OS version.

Interpreting Residuals

Once you have calculated your residuals, you can use them to assess the validity of your linear regression model. Here are some key considerations:

• Randomness: Ideally, the residuals should be randomly scattered around zero. A pattern in the residuals suggests that your linear model might not be appropriate for your data. For example, a curved pattern indicates a non-linear relationship.

• Magnitude: Large residuals indicate that your model isn't fitting those data points well. Investigate these points—they might represent outliers or data entry errors.

• Normal Distribution: The residuals should approximately follow a normal distribution (bell curve).

By carefully analyzing the residuals, you can gain valuable insights into the quality of your regression model and identify potential problems.

Further Analysis

Beyond simple linear regression, you can perform this process for other regression types available on your TI-84 (e.g., quadratic, cubic, exponential). The principle remains the same; the calculator will compute residuals based on the chosen model.

Remember to always visually inspect your residuals in a scatter plot. This provides a more intuitive understanding than just looking at the numerical values. This, combined with the numerical residuals, gives you a much more complete analysis of the relationship between variables.