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Linear Regression 3F4263

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Linear Regression 3F4263


1. **Stating the problem:** We are given data for the number of visitors ($x$) and daily sales ($y$), and we need to find the regression equation $\hat{y} = b_0 + b_1 x$. 2. **Formulas used:** - Slope: $$b_1 = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ - Intercept: $$b_0 = \bar{y} - b_1 \bar{x}$$ where $n$ is the number of data points, $\sum x$, $\sum y$, $\sum x^2$, and $\sum xy$ are sums of $x$, $y$, $x^2$, and $xy$ respectively. 3. **Calculate sums:** Calculate $\sum x$, $\sum y$, $\sum x^2$, and $\sum xy$ from the data table by squaring each $x$ and multiplying each $x$ by corresponding $y$, then summing. 4. **Calculate slope $b_1$:** Use the formula above with the sums. 5. **Calculate intercept $b_0$:** Calculate means $\bar{x} = \frac{\sum x}{n}$ and $\bar{y} = \frac{\sum y}{n}$, then find $b_0$. 6. **Regression equation:** Write the equation $\hat{y} = b_0 + b_1 x$. 7. **Prediction for $x=55$:** Calculate $\hat{y}$ by substituting $x=55$ into the regression equation. 8. **Conclusion:** Interpret $b_1$ as the expected increase in sales per additional visitor. State the predicted sales for 55 visitors. Since the exact sums are not provided, the user must calculate $\sum x$, $\sum y$, $\sum x^2$, and $\sum xy$ from their data to complete the solution.