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
sales (y) (in thousands of dollars)	8626	7623	6506	6527	4704	3632	4122	2215	2355	1143	571

Explanation:

Step1: Calculate means of x and y

Let \(x_i\) be the week numbers and \(y_i\) be the sales values.
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\), where \(n = 11\).
\(\sum_{i=1}^{11}x_i=1 + 2+\cdots+11=\frac{11\times(11 + 1)}{2}=66\), so \(\bar{x}=\frac{66}{11}=6\).
\(\sum_{i = 1}^{11}y_i=8626+7623+\cdots + 571=43388\), so \(\bar{y}=\frac{43388}{11}\approx3944.364\).

Step2: Calculate slope \(b_1\)

\(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=(1 - 6)^2+(2 - 6)^2+\cdots+(11 - 6)^2\)
\(=(- 5)^2+(-4)^2+\cdots+5^2=2\times(1^2 + 2^2+3^2+4^2+5^2)=2\times\frac{5\times(5 + 1)\times(2\times5+1)}{6}=110\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(1 - 6)(8626 - 3944.364)+(2 - 6)(7623 - 3944.364)+\cdots+(11 - 6)(571 - 3944.364)\)
After calculation, \(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=- 33799.54\)
\(b_1=\frac{-33799.54}{110}\approx - 307.27\)

Step3: Calculate intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 3944.364-(-307.27)\times6=3944.364 + 1843.62=5787.984\)
The linear - regression equation is \(\hat{y}=b_0 + b_1x=5787.984-307.27x\)

Step4: Calculate residuals

Residual \(e_i=y_i-\hat{y}_i\), for \(i = 1,2,\cdots,11\). For example, when \(x = 1\), \(\hat{y}_1=5787.984-307.27\times1=5480.714\), \(e_1=8626 - 5480.714 = 3145.286\)
Do this for all \(x\) values from \(1\) to \(11\).

Step5: Analyze residual plot

If the points in the residual plot are randomly scattered around the horizontal axis, the model is a good fit. If there is a pattern (such as a curve), the model is not a good fit.

Answer:

The linear - regression equation is \(\hat{y}=5787.984-307.27x\). Analyze the residual plot (by calculating residuals \(e_i=y_i-\hat{y}_i\) for \(i = 1,\cdots,11\)) to determine if the model is a good fit.