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Let's break this problem into parts and solve it step by step.

### Part (a): Finding the least squares regression line

We are given:
- Prices $x = \{3.25, 3.50, 3.75\}$
- Demand $y = \{4500, 3750, 3300\}$

We will calculate the least squares regression line in the form:
\[
y = mx + b
\]
Where $m$ is the slope and $b$ is the y-intercept.

The formulas for the slope $m$ and intercept $b$ are:
\[
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
\[
b = \frac{\sum y - m\sum x}{n}
\]
where $n$ is the number of data points (in this case, $n = 3$).

Let's calculate the necessary terms:
- $ \sum x = 3.25 + 3.50 + 3.75 $
- $ \sum y = 4500 + 3750 + 3300 $
- $ \sum xy = (3.25 \times 4500) + (3.50 \times 3750) + (3.75 \times 3300) $
- $ \sum x^2 = (3.25)^2 + (3.50)^2 + (3.75)^2 $

### Part (b): Estimating the demand for $x = 3.65$

Once we have the equation from part (a), we substitute $x = 3.65$ into the regression equation to estimate the demand.

Let me compute these values for you.### Solution:

#### Part (a): Least Squares Regression Line
The equation of the least squares regression line is:
\[
y = -2400x + 12250
\]
This means that for every unit increase in price, the demand decreases by 2400 gallons.

#### Part (b): Estimate of Demand when Price is $3.65
When the price is $3.65, the estimated demand is:
\[
y = -2400(3.65) + 12250 = 3490 \, \text{gallons}
\]

Would you like further details or explanations?

Here are 5 related questions to extend your understanding:
1. What is the significance of the slope in a linear regression model?
2. How can you calculate the correlation between price and demand?
3. How would you predict demand for prices outside the range provided?
4. What does a negative slope indicate in this real-world context?
5. How would the regression change if an outlier existed?

**Tip:** In regression, always check the goodness-of-fit (R² value) to understand how well the model predicts outcomes!

A fuel refiner wants to know the demand for a grade of gasoline as a function of price. The table shows daily sales y (in gallons) for three different prices: $3.25 (4500 gallons), $3.50 (3750 gallons), and $3.75 (3300 gallons). Find the least squares regression line and estimate the demand when the price is $3.65.

Learn how to calculate the least squares regression line for gasoline demand based on price. Given data points for prices ($3.25, $3.50, $3.75) and demand (4500, 3750, 3300 gallons), this problem demonstrates how to estimate demand when the price is $3.65. Suitable for advanced high school or college-level students.

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