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!