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Let's solve this problem step by step.

### Step 1: Understanding the data

We are given the following dataLet's solve this problem step by step.

### Step 1: Understanding the data

We are given the following 

| Year  | Selling Price $ p $ ($100) | Sales $ q $ (billions) |
|-------|------------------------------|--------------------------|
| 2012  | 4                            | 0.5                      |
| 2013  | 3                            | 1                        |
| 2017  | 2                            | 2                        |

We need to find the regression line equation for $ q(p) $, where $ q $ represents sales in billions and $ p $ is the price in hundreds of dollars.

### Step 2: Convert selling prices and sales into coordinate form

The data points are:
- (4, 0.5)
- (3, 1)
- (2, 2)

### Step 3: Calculate the slope of the regression line

The regression line is in the form:

\[
q(p) = mp + b
\]

Where $ m $ is the slope and $ b $ is the y-intercept. The formula for the slope is:

\[
m = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sum (x_i - \bar{x})^2 }
\]

Where:
- $ x_i $ are the selling prices (4, 3, 2),
- $ y_i $ are the sales values (0.5, 1, 2),
- $ \bar{x} $ is the mean of the selling prices,
- $ \bar{y} $ is the mean of the sales values.

#### Find $ \bar{x} $ and $ \bar{y} $:
\[
\bar{x} = \frac{4 + 3 + 2}{3} = 3
\]
\[
\bar{y} = \frac{0.5 + 1 + 2}{3} = 1.17
\]

#### Calculate $ m $ (slope):
\[
m = \frac{(4-3)(0.5-1.17) + (3-3)(1-1.17) + (2-3)(2-1.17)}{(4-3)^2 + (3-3)^2 + (2-3)^2}
\]

\[
m = \frac{(1)(-0.67) + (0)(-0.17) + (-1)(0.83)}{(1)^2 + (0)^2 + (-1)^2}
\]

\[
m = \frac{-0.67 - 0.83}{1 + 1} = \frac{-1.5}{2} = -0.75
\]

### Step 4: Find the y-intercept $ b $

We can now use the point-slope form to find $ b $:

\[
b = \bar{y} - m\bar{x}
\]

\[
b = 1.17 - (-0.75)(3) = 1.17 + 2.25 = 3.42
\]

### Step 5: Write the regression equation

The regression line equation is:

\[
q(p) = -0.75p + 3.4
\]

### Step 6: Use the regression line to estimate demand when price is $330

Given that the price is $330, we first convert it to hundreds of dollars:

\[
p = \frac{330}{100} = 3.3
\]

Now, substitute $ p = 3.3 $ into the regression equation:

\[
q(3.3) = -0.75(3.3) + 3.4
\]

\[
q(3.3) = -2.475 + 3.4 = 0.925
\]

So, when the selling price was $330, the estimated demand is approximately **0.93 billion** units (or **930 million** units).

---

Would you like further clarification on any step, or do you have additional questions?

### Related Questions:
1. How do you calculate the correlation coefficient for this data?
2. What does the slope of the regression line tell us about the relationship between price and sales?
3. Can we use the regression equation to predict sales if the price drops to $150?
4. How reliable is this regression line for future predictions?
5. How would the equation change if a new data point for 2020 were added?

#### Tip:
Always check the validity of your regression model by examining how well it fits the data, especially if you're using it for predictions outside the observed range.

The table shows selling prices and sales of smartphones in 2012, 2013, and 2017. Find the regression line and estimate the demand when the selling price is $330.

Learn how to find the regression line for smartphone sales data from 2012, 2013, and 2017, and use it to estimate demand when the selling price is $330.

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