To develop the estimated regression equation for these data, we need to find the slope $b_1$ and the y-intercept $b_0$ using the following formulas:

1. Slope $b_1$

The formula for $b_1$ (the slope of the regression line) is:

$b_1 = \frac{n \sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}$

2. Intercept $b_0$

The formula for $b_0$ (the intercept of the regression line) is:

$b_0 = \frac{\sum{y_i} - b_1 \sum{x_i}}{n}$

Where:

• $n$ is the number of data points
• $x_i$ and $y_i$ are the data points for the variables $x$ and $y$

Let’s calculate these values step by step.

Given Data:

$x_i: 6, 11, 15, 18, 20$ $y_i: 6, 8, 12, 20, 30$

Step 1: Compute the necessary sums

• $\sum{x_i} = 6 + 11 + 15 + 18 + 20 = 70$
• $\sum{y_i} = 6 + 8 + 12 + 20 + 30 = 76$
• $\sum{x_i^2} = 6^2 + 11^2 + 15^2 + 18^2 + 20^2 = 36 + 121 + 225 + 324 + 400 = 1106$
• $\sum{x_i y_i} = 6(6) + 11(8) + 15(12) + 18(20) + 20(30) = 36 + 88 + 180 + 360 + 600 = 1264$

Step 2: Calculate the slope $b_1$

Using the formula for $b_1$:

$b_1 = \frac{5(1264) - (70)(76)}{5(1106) - (70)^2}$ $b_1 = \frac{6320 - 5320}{5530 - 4900}$ $b_1 = \frac{1000}{630} \approx 1.587$

Step 3: Calculate the intercept $b_0$

Now we can compute $b_0$ using the formula:

$b_0 = \frac{76 - (1.587)(70)}{5}$ $b_0 = \frac{76 - 111.09}{5}$ $b_0 = \frac{-35.09}{5} \approx -7.018$

Final Estimated Regression Equation:

$\hat{y} = -7.018 + 1.587x$

So the estimated regression equation is:

$\hat{y} = -7.018 + 1.587x$

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Would you like more details or clarifications on any of these steps? Here are some follow-up questions that might interest you:

1. How can we interpret the slope and intercept of the regression equation in real-life terms?
2. What is the method to assess the accuracy of the regression equation?
3. How can we compute the correlation coefficient from the regression equation?
4. How would adding more data points affect the regression equation?
5. What assumptions are made when applying linear regression?

Tip: When working with regression, always check for outliers or unusual data points that could skew your results!