Question

the data below represents an international corporations internal estimates of sales (in thousands of dollars) in the coming year over time (in weeks). use a linear regression to model the data. round all your coefficients to three decimal places. then use a residual plot to determine if your model is a good fit.

week (x)	1	2	3	4	5	6	7	8	9	10	11

Explanation:

Step1: Calculate sums

Let $n = 11$. Calculate $\sum_{i = 1}^{n}x_i=1 + 2+\cdots+11=\frac{11\times(11 + 1)}{2}=66$, $\sum_{i = 1}^{n}y_i=1275+2635+\cdots+2115650$, $\sum_{i = 1}^{n}x_i^2=1^2+2^2+\cdots+11^2=\frac{11\times(11 + 1)\times(2\times11 + 1)}{6}=506$, and $\sum_{i = 1}^{n}x_iy_i$.

Step2: Find slope $m$

The formula for the slope $m$ of the least - squares regression line $y=mx + b$ is $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$. Substitute the sums calculated in Step 1 to find $m$.

Step3: Find intercept $b$

The formula for the intercept $b$ is $b=\overline{y}-m\overline{x}$, where $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$.

Step4: Calculate residuals

For each data - point $(x_i,y_i)$, calculate the residual $e_i=y_i-(mx_i + b)$.

Step5: Create residual plot

Plot the residuals $e_i$ against the $x$ - values. If the points are randomly scattered around the horizontal axis, the linear regression model is a good fit.

Answer:

The steps above outline the process to find the linear regression model and use a residual plot to determine if it is a good fit. The actual values of $m$, $b$ and the conclusion about the goodness - of - fit require the numerical sums from Step 1 to be calculated precisely. After calculating the sums:

1. Calculate $m$ using the formula.
2. Calculate $b$ using the formula.
3. Calculate the residuals for each data - point.
4. Plot the residuals against $x$ values and analyze the pattern of the points on the plot. If the points are randomly scattered, the model is a good fit; if there is a pattern (e.g., a curve), the model is not a good fit.