Introduction

This document guides you through performing Simple Linear Regression (SLR) using a TI-84 calculator, showing step-by-step instructions, output interpretation, and a practical example.

Step-by-Step Guide on TI-84

1. Entering Data

  1. Turn on your calculator and press STAT.
  2. Choose 1: Edit and press ENTER.
  3. You will see columns labeled L1, L2, etc.
    • Enter your x values (independent variable) into L1.
    • Enter your y values (dependent variable) into L2.
  4. If there is existing data in L1 or L2, you can clear it by navigating to the column name (e.g., L1), pressing CLEAR, and then ENTER.

2. Running the Linear Regression Calculation

  1. Press STAT, use the arrow keys to navigate to CALC, and choose 4: LinReg(ax + b).

  2. You should see LinReg(ax+b) appear on the screen. If it does not, press 4 after navigating to CALC.

  3. Next, you need to specify the lists where your data is stored. If your x values are in L1 and your y values are in L2, enter:

    LinReg(ax+b) L1, L2

    Press 2ND then 1 to insert L1, and press 2ND then 2 to insert L2.

  4. Press ENTER. The calculator will display the regression output.

Interpreting the Output

The calculator will show the following results: - a: The slope (\(\hat{eta_1}\)) of the regression line. - b: The intercept (\(\hat{eta_0}\)) of the regression line. - r: The correlation coefficient, showing the strength and direction of the linear relationship. - r²: The coefficient of determination (\(R^2\)), indicating how well the regression line explains the data.

Example:

Suppose you have the following data:

L1 1 2 3 4 5 6
L2 2 4 5 4 5 7
  1. Enter these x values into L1 and y values into L2.
  2. Run LinReg(ax+b) L1, L2.

Expected Output:

a = 0.7857
b = 1.9048
r = 0.8995
r² = 0.8091

Output Interpretation

  1. Equation of the Line: The equation of the regression line is: \[ \hat{y} = ax + b \] Substituting the values: \[ \hat{y} = 0.7857x + 1.9048 \]

  2. Slope (a = 0.7857):

    • The slope tells us that for each unit increase in x, the y value is expected to increase by approximately 0.7857.
    • Interpretation: In this case, if x increases by 1, y will increase by about 0.7857 units on average.

  3. Intercept (b = 1.9048):

    • The intercept is the expected value of y when x is 0.
    • Interpretation: If x = 0, the predicted y value would be 1.9048.

  4. Correlation Coefficient (r = 0.8995):

    • The correlation coefficient shows the strength and direction of the linear relationship.
    • Since r is close to 1, it indicates a strong positive linear relationship between x and y.

  5. Coefficient of Determination (\(R^2 = 0.8091\)):

    • \(R^2\) represents the proportion of the variance in y that can be explained by the linear relationship with x.
    • Interpretation: In this example, approximately 80.91% of the variability in y is explained by the linear relationship with x.

Step-by-Step Example on Prediction

  1. Suppose you want to predict the y value for x = 7.
  2. Use the regression equation: \[ \hat{y} = 0.7857 \cdot 7 + 1.9048 \]
  3. Perform the calculation: \[ \hat{y} = 5.4999 + 1.9048 = 7.4047 \]
  4. Interpretation: When x = 7, the predicted y value is approximately 7.4.

Tips:

  • Ensure your x and y lists have the same number of data points.
  • If the r value does not show up, you may need to enable the Diagnostic feature:
    1. Press 2ND then 0 (for CATALOG).
    2. Scroll to DiagnosticOn and press ENTER twice.

Following these steps and understanding the output will allow you to perform and interpret Simple Linear Regression using the TI-84 calculator effectively.