Dr. Lillian​ Fok, a New Orleans​ psychologist, specializes in treating patients who are agoraphobic​ (i.e., afraid to leave their​ homes). The following table indicates how many patients Dr. Fok has seen each year for the past 10 years. It also indicates what the robbery rate was in New Orleans during the same​ year:
                                                                                                                                                                     
Year
1
2
3
4
5
6
7
8
9
10
Number of Patients
37
34
39
40
41
55
62
55
58
60
Robbery Rate per​ 1,000 Population
57.8
61.4
72.6
75.7
81.1
88.6
101.1
94.2
104.1
117.2
 
Part 2
The simple linear regression equation that shows the best relationship between the number of patients and year is ​(round your responses to three decimal​ places):
                                               
ModifyingAbove y with caret ​= 
  
enter your response here ​+ 
  
enter your response herex
where ModifyingAbove y with caret ​= Dependent Variable and x​ = Independent Variable.

To find the simple linear regression equation between the number of patients (dependent variable, \( y \)) and the year (independent variable, \( x \)), we can follow these steps:

### Step 1: Organize the Data
The data provided for each year (1 to 10) and the corresponding number of patients is as follows:

| Year (\( x \)) | Number of Patients (\( y \)) |
| ---------------|-----------------------------|
| 1              | 37                          |
| 2              | 34                          |
| 3              | 39                          |
| 4              | 40                          |
| 5              | 41                          |
| 6              | 55                          |
| 7              | 62                          |
| 8              | 55                          |
| 9              | 58                          |
| 10             | 60                          |

### Step 2: Use the Formula for Simple Linear Regression
The linear regression equation is given by:

\[
y = b_0 + b_1x
\]

Where:
- \( b_0 \) is the y-intercept
- \( b_1 \) is the slope of the line

To calculate \( b_0 \) and \( b_1 \), we use the following formulas:
\[
b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
\[
b_0 = \frac{\sum y - b_1 \sum x}{n}
\]

Where:
- \( n \) is the number of data points
- \( \sum xy \) is the sum of the product of corresponding values of \( x \) and \( y \)
- \( \sum x \), \( \sum y \), and \( \sum x^2 \) are the sums of the values of \( x \), \( y \), and the squares of \( x \), respectively.

### Step 3: Perform the Calculations
I'll calculate these values now.The simple linear regression equation that best fits the data is:

\[
\hat{y} = 30.267 + 3.242x
\]

Where:
- \( \hat{y} \) represents the predicted number of patients
- \( x \) represents the year.

Would you like to explore any other details related to this regression, or have any further questions?

Here are 5 related questions to consider:
1. How do the residuals (errors) look for this regression model?
2. What does the slope of 3.242 mean in the context of this problem?
3. Can we predict the number of patients for year 11 using this model?
4. How strong is the relationship between the number of patients and year?
5. What would happen if we included robbery rates as another variable in the model?

**Tip:** Always check the residuals in a regression model to ensure it fits the data well before making predictions.

Dr. Lillian Fok, a New Orleans psychologist, specializes in treating patients who are agoraphobic (i.e., afraid to leave their homes). The following table indicates how many patients Dr. Fok has seen each year for the past 10 years. It also indicates what the robbery rate was in New Orleans during the same year.

Explore the simple linear regression model showing the relationship between the number of agoraphobia patients treated by Dr. Lillian Fok over 10 years and the corresponding yearly trends. Includes calculations using the least squares method and detailed steps for finding the regression equation.

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