Question 1163400: For a sample of 29 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $1,000s). He finds that SSE = 4,136,781 and SST = 7,662,937.
Calculate the standard error of the estimate.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! To calculate the standard error of the estimate ($S_e$) for a multiple regression model, we use the sum of squared errors ($SSE$) and the degrees of freedom associated with the error.
### 1. Identify the Given Information
* **Sample size ($n$):** $29$
* **Number of independent variables ($k$):** $2$ (Poverty rate and Median income)
* **Sum of Squared Errors ($SSE$):** $4,136,781$
* **Total Sum of Squares ($SST$):** $7,662,937$ (Note: $SST$ is not required for this specific calculation)
### 2. The Formula
The standard error of the estimate is the square root of the mean square error ($MSE$):
$$S_e = \sqrt{\frac{SSE}{n - k - 1}}$$
Where:
* $n - k - 1$ represents the degrees of freedom for the error ($df_E$).
### 3. Calculation
**Step 1: Calculate the degrees of freedom ($df_E$)**
$$df_E = 29 - 2 - 1 = 26$$
**Step 2: Calculate the Mean Square Error ($MSE$)**
$$MSE = \frac{4,136,781}{26} \approx 159,106.9615$$
**Step 3: Calculate the square root**
$$S_e = \sqrt{159,106.9615} \approx 398.8821$$
### Final Answer
The standard error of the estimate is approximately **$398.88$**.
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