A shoe store developed the following estimated regression equation relating sales to inventory investment and advertising expenditures. ŷ = 25 + 12xi + 6x2 where T1 = inventory investment ($1000s) T2 = advertising expenditures ($1000s) y = sales ($1000s) a. Predict the sales resulting from a $15,000 investment in inventory and an advertising budget of $10,000.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
a. Predict the sales resulting from a $15,000 investment in inventory and an advertising budget of $10,000.
![**Regression Equation Analysis**
A shoe store has developed the following estimated regression equation to relate sales to inventory investment and advertising expenditures:
\[ \hat{y} = 25 + 12x_1 + 6x_2 \]
**Where:**
- \( x_1 \) = inventory investment ($1000s)
- \( x_2 \) = advertising expenditures ($1000s)
- \(\hat{y}\) = sales ($1000s)
**Tasks:**
a. **Predict Sales**
- For a $15,000 investment in inventory and an advertising budget of $10,000, predict sales using the equation.
b. **Interpretation of \( b_1 \) and \( b_2 \)**
- **\( b_1 \)**: Indicates how sales can be expected to change with each $1000 increase in inventory investment, holding advertising constant.
- **\( b_2 \)**: Indicates how sales can be expected to change with each $1000 increase in advertising expenditure, holding inventory investment constant.
**Interactive Components:**
- **Select Options**: Engage with dropdown selections to apply changes and understand impacts on sales predictions with varying investments in inventory or advertising, while keeping other variables constant.
- **Check My Work**: Use this feature to verify the correct application of the regression equation.
**Exercise Context:**
This exercise (Exercise 15.04 Algo) uses the Least Squares Method to provide practical regression analysis, enhancing understanding of multiple regression in business settings.
> Instructions for Educators: Guide students through the equation's application, emphasizing the role of coefficients and predictions. Encourage exploration through interactive options to solidify understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc8e2d68-695c-49ad-a9b1-8d66f1d1c9ea%2F8e9814f5-2894-4306-b7ca-a06bd6cd3d9c%2Fsyob7zt_processed.jpeg&w=3840&q=75)
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