​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**

​$2.502.50

770770

​$3.503.50

505505

​$2.102.10

990990

​$4.204.20

245245

​$3.203.20

315315

​$4.104.10

475475

Part 2

Using

simple linear regression

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and given that the price per cup is

​$1.851.85​,

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 for mocha latte coffees based on the given prices and quantities sold, we can use **simple linear regression**. The formula for a simple linear regression model is:

\[
y = a + bx
\]

Where:
- $ y $ is the dependent variable (demand or number of cups sold).
- $ x $ is the independent variable (price per cup).
- $ a $ is the intercept.
- $ b $ is the slope of the regression line.

We'll first calculate the values for $ a $ and $ b $ using the data provided. The data points are:

| Price $ x $ | Number Sold $ y $ |
|---------------|--------------------|
| 2.50          | 770                |
| 3.50          | 505                |
| 2.10          | 990                |
| 4.20          | 245                |
| 3.20          | 315                |
| 4.10          | 475                |

We need to find the slope $ b $ and intercept $ a $ using these formulas:

\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
a = \frac{\sum y - b(\sum x)}{n}
\]

Where:
- $ n $ is the number of data points (in this case, 6).
- $ \sum xy $ is the sum of the products of $ x $ and $ y $.
- $ \sum x $ is the sum of the prices.
- $ \sum y $ is the sum of the quantities sold.
- $ \sum x^2 $ is the sum of the squares of the prices.

Let me calculate these values to find the forecast.The forecasted demand for mocha latte coffees, when the price per cup is set at $1.85, is **956 cups** (rounded to one decimal place).

Do you want more details or have any questions?

Here are five related questions to consider:
1. How does changing the price affect the forecasted demand in general?
2. What would the forecast be if the price was higher, such as $5.00?
3. Can this model be used for predicting demand at extremely low or high prices?
4. How accurate is simple linear regression for this dataset?
5. What is the significance of the slope in this context?

**Tip:** Always check the correlation between variables to ensure that linear regression is appropriate for your data!

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: $2.50, Number Sold: 770; Price: $3.50, Number Sold: 505; Price: $2.10, Number Sold: 990; Price: $4.20, Number Sold: 245; Price: $3.20, Number Sold: 315; Price: $4.10, Number Sold: 475.

Using simple linear regression and given that the price per cup is $1.85, what is the forecasted demand for mocha latte coffees?

Learn how to forecast demand for mocha latte coffees using simple linear regression based on historical price and sales data. This problem demonstrates key concepts in linear regression, including calculating slope and intercept. Forecast the demand when the price per cup is $1.85. Suitable for students in Grades 10-12.

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