Weight (pounds), x	Miles per Gallon, y

3723	18

3767	17

2739	25

3490	19

3242	22

2935	23

3764	16

2630	24

3400	18

3707	17 Find the​ least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable.

ModifyingAbove y with caretyequals=enter your response herexplus+​(enter your response here​)

​(Round the x coefficient to five decimal places as needed. Round the constant to one decimal place as​ needed.)

To find the least-squares regression line, we can use the formula for a simple linear regression model, which is:

\[
\hat{y} = b_1 x + b_0
\]

Where:
- \( \hat{y} \) is the predicted value of the dependent variable (Miles per Gallon),
- \( x \) is the independent variable (Weight),
- \( b_1 \) is the slope of the regression line,
- \( b_0 \) is the y-intercept.

We need to calculate:
1. The slope (\( b_1 \)) using the formula:
   \[
   b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
   \]
   
2. The intercept (\( b_0 \)) using the formula:
   \[
   b_0 = \bar{y} - b_1 \bar{x}
   \]
   where \( \bar{x} \) and \( \bar{y} \) are the means of \( x \) and \( y \).

Let me calculate this for you.The least-squares regression line is:

\[
\hat{y} = -0.00724x + 44.1
\]

Here, the slope is \(-0.00724\) (rounded to five decimal places), and the intercept is \(44.1\) (rounded to one decimal place).

Would you like any further details or explanations?

Here are five related questions for practice:
1. What does the slope of \(-0.00724\) tell us about the relationship between weight and miles per gallon?
2. How would you predict the miles per gallon for a car weighing 3000 pounds?
3. How does an increase in weight affect the fuel efficiency, based on this model?
4. What is the correlation coefficient for this data set?
5. Can this model be used to predict the miles per gallon for cars significantly lighter or heavier than those in the data set?

**Tip:** The slope indicates how much the dependent variable (miles per gallon) changes for each unit change in the independent variable (weight).

Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. Data: (Weight (pounds), Miles per Gallon): [3723, 18], [3767, 17], [2739, 25], [3490, 19], [3242, 22], [2935, 23], [3764, 16], [2630, 24], [3400, 18], [3707, 17]. Round the slope to five decimal places and the intercept to one decimal place.

Learn how to calculate the least-squares regression line using weight as the explanatory variable and miles per gallon as the response. Includes step-by-step calculations and key formulas for linear regression, suitable for students in Grades 10-12.

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