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Part 1
The following data relate the sales figures of the bar in Mark​ Kaltenbach's small​ bed-and-breakfast inn in​ Portland, to the number of guests registered that​ week:
                                                                                           
Week
Guests
Bar Sales
1
16
​$330
2
12
​$265
3
18
​$380
4
14
​$315
 
Part 2
​a) The simple linear regressionLOADING... equation that relates bar sales to number of guests​ (not to​ time) is ​(round your responses to one decimal​ place):
                                               
Bar Sales​ = 
  
enter your response here ​+ 
  
enter your response heretimesguests
Part 3
​b) If the forecast is 24 guests next​ week, the bar sales are expected to be ​$
  
enter your response here ​(round your response to one decimal​ place).

### Part 2: Simple Linear Regression

To determine the simple linear regression equation, we will use the formula for a linear regression model:

\[
\hat{y} = b_0 + b_1x
\]

Where:
- $\hat{y}$ is the predicted value (Bar Sales in this case),
- $x$ is the independent variable (Guests),
- $b_0$ is the intercept (the value of $\hat{y}$ when $x = 0$),
- $b_1$ is the slope of the regression line (which represents the change in Bar Sales per additional guest).

To calculate this, we'll need to use the following formulas:
- The slope ($b_1$) is given by:
\[
b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
- The intercept ($b_0$) is given by:
\[
b_0 = \frac{\sum y - b_1(\sum x)}{n}
\]

Where:
- $n$ is the number of observations,
- $x$ is the number of guests,
- $y$ is the bar sales.

We will first calculate all the necessary sums, then compute the slope and intercept.

#### Data:
- Guests: $x = [16, 12, 18, 14]$
- Bar Sales: $y = [330, 265, 380, 315]$

Let's compute the values.The simple linear regression equation that relates bar sales to the number of guests is:

\[
\text{Bar Sales} = 52.5 + 18.0 \times \text{Guests}
\]

### Part 3: Forecast for 24 Guests

To forecast bar sales for 24 guests, we substitute $x = 24$ into the regression equation:

\[
\hat{y} = 52.5 + 18.0 \times 24
\]

Let's calculate the expected bar sales for 24 guests.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the forecasted sales by plugging $ x = 24 $ into the equation:

\[
\text{Bar Sales} = 52.5 + 18.0 \times 24
\]

Let me know if you'd like further clarification or help with this!

### Questions to expand your understanding:
1. What factors can affect the slope in a regression model?
2. How does the intercept change when the data is scaled up or down?
3. Can linear regression be applied when the relationship between variables is not linear?
4. What other methods can you use to predict bar sales if guest data follows a non-linear pattern?
5. How can you assess the accuracy of your linear regression model?

**Tip**: Always plot your data before running a regression to visually inspect if the relationship looks linear!

The following data relate the sales figures of the bar in Mark Kaltenbach's small bed-and-breakfast inn in Portland, to the number of guests registered that week:

Week  | Guests | Bar Sales
1     | 16     | $330
2     | 12     | $265
3     | 18     | $380
4     | 14     | $315

Part 2: a) The simple linear regression equation that relates bar sales to number of guests (round your responses to one decimal place):
Bar Sales = _____ + _____ times guests

Part 3: b) If the forecast is 24 guests next week, the bar sales are expected to be $_____ (round your response to one decimal place).

Learn how to calculate the simple linear regression equation for predicting bar sales based on the number of guests at a bed-and-breakfast in Portland. This example includes step-by-step use of key formulas such as slope, intercept, and least squares method. Suitable for students in Grades 10-12.

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