To calculate the standardized coefficients for predictors $A$ (denoted as $\beta_A$) and $B$ (denoted as $\beta_B$), we will need to transform the unstandardized coefficients to their standardized form using the following formula:

$\beta_{\text{std}} = \beta_{\text{unstd}} \times \frac{\text{SD of X}}{\text{SD of Y}}$

Where:

• $\beta_{\text{std}}$ is the standardized coefficient.
• $\beta_{\text{unstd}}$ is the unstandardized coefficient.
• SD of X is the standard deviation of the predictor.
• SD of Y is the standard deviation of the dependent variable (Y).

We are given the following information:

• Unstandardized coefficients:

  • $\beta_A = 0.2002379$
  • $\beta_B = 1.0040400$

• Standard deviations:

  • SD of $A$ = 5.72
  • SD of $B$ = 1.01
  • We need to calculate SD of $Y$ using the formula derived from the $R^2$ value and the correlation coefficients.

Step 1: Calculate SD of Y

To calculate the SD of $Y$, we use the following relationship from the regression model:

$\text{R}^2 = r_{y1}^2 + r_{y2}^2 + 2r_{y1}r_{y2}r_{12}$

Where:

• $r_{y1}$ is the correlation between $X_1$ and $Y$,
• $r_{y2}$ is the correlation between $X_2$ and $Y$,
• $r_{12}$ is the correlation between $X_1$ and $X_2$.

Using this, we can estimate the SD of $Y$ as follows: $\text{SD of Y} = \frac{\text{residual\_se}}{\sqrt{1 - \text{R}^2}}$

Step 2: Compute Standardized Coefficients

Once we have the SD of $Y$, we can calculate the standardized coefficients for $A$ and $B$ using the formula provided above.

Let me do the calculations now. math

Given values

r_y1 = 0.07982481 # Correlation between X1 and Y r_y2 = 0.25408597 # Correlation between X2 and Y r_12 = 0.04279626 # Correlation between X1 and X2

residual_se = 0.09105 # Residual standard error R_squared = 0.9964 # R-squared value

sd_X_A = 5.72 # Standard deviation of predictor A sd_X_B = 1.01 # Standard deviation of predictor B

beta_A_unstd = 0.2002379 # Unstandardized coefficient for A beta_B_unstd = 1.0040400 # Unstandardized coefficient for B

Calculate SD of Y using the formula: SD_Y = residual_se / sqrt(1 - R_squared)

SD_Y = residual_se / math.sqrt(1 - R_squared)

Standardized coefficients formula:

beta_std = beta_unstd * (SD_X / SD_Y)

beta_A_std = beta_A_unstd * (sd_X_A / SD_Y) beta_B_std = beta_B_unstd * (sd_X_B / SD_Y)

beta_A_std, beta_B_stdThe standardized coefficients for the predictors are:

• Standardized coefficient for A: $\beta_A = 0.7548$
• Standardized coefficient for B: $\beta_B = 0.6683$

Would you like further details on the calculations, or do you have any other questions?

Here are some related questions to explore further:

1. How does the $R^2$ value influence the interpretation of regression results?
2. What is the significance of standardizing coefficients in regression models?
3. How do residual standard errors impact the reliability of a regression model?
4. How can we interpret the correlation between predictors in multiple regression?
5. What role do standard deviations play in regression analysis?

Tip: Always ensure that the predictors have been scaled appropriately before comparing standardized coefficients, especially when they have different units of measurement.