Answer :
Certainly! Let's dive into a detailed analysis of the given data.
The table shows the relationship between the amount invested in advertising and the expected yearly profit for a company. Here's how you can understand this relationship:
1. Data Interpretation:
- We are given five data points where each point is a pair: ([tex]\(x\)[/tex], [tex]\(y\)[/tex]).
- [tex]\(x\)[/tex] represents the amount invested in advertising (in hundreds of thousands of dollars).
- [tex]\(y\)[/tex] represents the yearly profit (also in hundreds of thousands of dollars).
Data points:
- (1, 28.8)
- (3, 36.8)
- (5, 40.7)
- (7, 37.9)
- (9, 32.6)
2. Finding the Correlation Coefficient:
- The correlation coefficient helps us understand the strength and direction of the linear relationship between the investment in advertising and the yearly profit.
- The value of the correlation coefficient is approximately 0.2938.
- This value indicates a weak positive linear relationship. In other words, as the investment in advertising increases, the profit tends to increase slightly, but not strongly.
3. Linear Regression Analysis:
- We perform linear regression to fit a straight line (a linear model) to the data points. This model predicts the profit based on the amount invested in advertising.
- The linear equation derived from this is: [tex]\( y = 0.435x + 33.185 \)[/tex].
- Slope (0.435): Indicates that for each additional hundred thousand dollars spent on advertising, the yearly profit is expected to increase by 0.435 hundred thousand dollars (or $43,500).
- Intercept (33.185): Represents the estimated profit when no money is spent on advertising. This is the point where the line crosses the y-axis.
In summary, the analysis suggests that there's a weak positive correlation between advertising investment and profit, and the linear model provides a simple way to estimate potential profit based on different levels of investment.
The table shows the relationship between the amount invested in advertising and the expected yearly profit for a company. Here's how you can understand this relationship:
1. Data Interpretation:
- We are given five data points where each point is a pair: ([tex]\(x\)[/tex], [tex]\(y\)[/tex]).
- [tex]\(x\)[/tex] represents the amount invested in advertising (in hundreds of thousands of dollars).
- [tex]\(y\)[/tex] represents the yearly profit (also in hundreds of thousands of dollars).
Data points:
- (1, 28.8)
- (3, 36.8)
- (5, 40.7)
- (7, 37.9)
- (9, 32.6)
2. Finding the Correlation Coefficient:
- The correlation coefficient helps us understand the strength and direction of the linear relationship between the investment in advertising and the yearly profit.
- The value of the correlation coefficient is approximately 0.2938.
- This value indicates a weak positive linear relationship. In other words, as the investment in advertising increases, the profit tends to increase slightly, but not strongly.
3. Linear Regression Analysis:
- We perform linear regression to fit a straight line (a linear model) to the data points. This model predicts the profit based on the amount invested in advertising.
- The linear equation derived from this is: [tex]\( y = 0.435x + 33.185 \)[/tex].
- Slope (0.435): Indicates that for each additional hundred thousand dollars spent on advertising, the yearly profit is expected to increase by 0.435 hundred thousand dollars (or $43,500).
- Intercept (33.185): Represents the estimated profit when no money is spent on advertising. This is the point where the line crosses the y-axis.
In summary, the analysis suggests that there's a weak positive correlation between advertising investment and profit, and the linear model provides a simple way to estimate potential profit based on different levels of investment.
