Use the following census data from the cities of Washington and Franklin to answer the following questions.

Year

Washington

Population

​(in thousands)

Year

Franklin

Population

​(in thousands)

19801980

35.635.6

19801980

59.459.4

19901990

40.340.3

19901990

64.764.7

20002000

58.658.6

20002000

8484

20102010

74.474.4

20102010

87.587.5

20202020

85.185.1

20202020

94.194.1

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Part 1

**a.** Find the regression line and correlation coefficient for each set of data. Let x be the year.

Find the regression line for Washington population.

The regression line is

Yequals=enter your response here.

​(Use integers or decimals for any numbers in the expression. Round to four decimal places as​ needed.)

To find the regression line for the population of Washington, we'll use the least squares method to determine the line of best fit. The equation of the regression line is typically expressed as:

\[
Y = mx + b
\]

where:
- \( Y \) is the dependent variable (population in this case).
- \( x \) is the independent variable (year in this case).
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 1: Convert Years to a More Manageable Scale
Since the years are large numbers, we can scale them by subtracting 1980 from each year to make the computations easier.

Let's convert the years to:

\[
x = \text{Year} - 1980
\]

This gives us:

| Year | x (Year - 1980) | Washington Population (Y) |
|------|-----------------|---------------------------|
| 1980 | 0               | 35.6                      |
| 1990 | 10              | 40.3                      |
| 2000 | 20              | 58.6                      |
| 2010 | 30              | 74.4                      |
| 2020 | 40              | 85.1                      |

### Step 2: Calculate the Necessary Sums
We will now calculate the sums required to find the slope \( m \) and intercept \( b \).

\[
\text{Sum of } x = \sum{x}
\]
\[
\text{Sum of } Y = \sum{Y}
\]
\[
\text{Sum of } x^2 = \sum{x^2}
\]
\[
\text{Sum of } xY = \sum{xY}
\]

Let's calculate these:

\[
\sum{x} = 0 + 10 + 20 + 30 + 40 = 100
\]
\[
\sum{Y} = 35.6 + 40.3 + 58.6 + 74.4 + 85.1 = 294
\]
\[
\sum{x^2} = 0^2 + 10^2 + 20^2 + 30^2 + 40^2 = 3000
\]
\[
\sum{xY} = (0 \times 35.6) + (10 \times 40.3) + (20 \times 58.6) + (30 \times 74.4) + (40 \times 85.1) = 7802
\]

### Step 3: Calculate Slope \( m \) and Intercept \( b \)

\[
m = \frac{n(\sum{xY}) - (\sum{x})(\sum{Y})}{n(\sum{x^2}) - (\sum{x})^2}
\]

\[
m = \frac{5(7802) - (100)(294)}{5(3000) - 100^2} = \frac{39010 - 29400}{15000 - 10000} = \frac{9610}{5000} = 1.922
\]

\[
b = \frac{(\sum{Y}) - m(\sum{x})}{n}
\]

\[
b = \frac{294 - 1.922(100)}{5} = \frac{294 - 192.2}{5} = \frac{101.8}{5} = 20.36
\]

### Step 4: Write the Regression Line Equation
The regression line equation for Washington is:

\[
Y = 1.922x + 20.3600
\]

### Part 1a Answer:
The regression line for the population of Washington is:

\[
Y = 1.922x + 20.3600
\]

### Correlation Coefficient
The correlation coefficient (r) can be calculated using the formula:

\[
r = \frac{n(\sum{xY}) - (\sum{x})(\sum{Y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{Y^2} - (\sum{Y})^2]}}
\]

Would you like to proceed with calculating the correlation coefficient, or do you have any questions about this part?

Learn how to calculate the regression line and correlation coefficient for Washington population data using statistical methods. Step-by-step solution includes converting years, calculating sums, and deriving equations. Suitable for college-level students and statistics enthusiasts.

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