​Café Michigan's​ manager, Gary​ Stark, suspects that demand for mocha latte coffees depends on the price being charged.  Based on historical​ observations, Gary has gathered the following​ data, which show the numbers of these coffees sold over six different price​ values:
                                                                             
Price
Number Sold
​$2.70
765
​$3.40
510
​$2.00
975
​$4.10
245
​$3.10
320
​$4.00
490
 
Part 2
Using simple linear regressionLOADING... and given that the price per cup is ​$1.80​, the forecasted demand for mocha latte coffees will be 
  
enter your response here cups ​(enter your response rounded to one decimal​ place).

To forecast the demand using simple linear regression, we first need to calculate the linear regression equation in the form:

\[
y = mx + b
\]

where:
- $ y $ is the number of cups sold (dependent variable).
- $ x $ is the price (independent variable).
- $ m $ is the slope of the regression line.
- $ b $ is the y-intercept.

### Step 1: Organize the Data
Given dataTo forecast the demand using simple linear regression, we first need to calculate the linear regression equation in the form:

\[
y = mx + b
\]

where:
- $ y $ is the number of cups sold (dependent variable).
- $ x $ is the price (independent variable).
- $ m $ is the slope of the regression line.
- $ b $ is the y-intercept.

### Step 1: Organize the Data
Given 

| Price ($x$) | Number Sold ($y$) |
|---------------|---------------------|
| 2.70          | 765                 |
| 3.40          | 510                 |
| 2.00          | 975                 |
| 4.10          | 245                 |
| 3.10          | 320                 |
| 4.00          | 490                 |

### Step 2: Calculate the slope (m) and intercept (b)
We use the following formulas for the slope and intercept:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Where:
- $ n $ is the number of data points.
- $ \sum xy $ is the sum of the product of corresponding $ x $ and $ y $ values.
- $ \sum x $ is the sum of the $ x $ values.
- $ \sum y $ is the sum of the $ y $ values.
- $ \sum x^2 $ is the sum of the squares of the $ x $ values.

### Step 3: Apply Regression Equation to Predict Demand
Once we have the regression equation $ y = mx + b $, we can substitute $ x = 1.80 $ (the price) into the equation to forecast the demand.

Let me calculate the values for you:The forecasted demand for mocha latte coffees, given a price of $1.80 per cup, is approximately **961.9 cups** (rounded to one decimal place).

Would you like a detailed explanation of the calculations, or do you have any questions?

Here are 5 related questions that expand on this topic:
1. How would changing the price to $5.00 affect the forecasted demand?
2. What is the significance of the slope in a linear regression model?
3. How can we measure the accuracy of this linear regression forecast?
4. What would happen to the forecast if we included more historical data points?
5. How can other factors (besides price) be included in the demand model?

**Tip:** Linear regression is best suited when there is a linear relationship between the variables. Always visualize your data to confirm this before applying it.

Café Michigan's manager, Gary Stark, suspects that demand for mocha latte coffees depends on the price being charged. Based on historical observations, Gary has gathered the following data, which show the numbers of these coffees sold over six different price values:

Price | Number Sold
$2.70 | 765
$3.40 | 510
$2.00 | 975
$4.10 | 245
$3.10 | 320
$4.00 | 490

Using simple linear regression, and given that the price per cup is $1.80, the forecasted demand for mocha latte coffees will be ___ cups (enter your response rounded to one decimal place).

Learn how to forecast demand for mocha latte coffees using linear regression with historical data from Café Michigan. This detailed example covers key statistical concepts, including calculating the slope and intercept of a linear regression model. Suitable for students in Grades 10-12.

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