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)\tsales (y) (in thousands of dollars)
1\t26340
2\t26486
3\t26372
4\t25738
5\t25102
6\t23783
7\t22104
8\t20008
9\t17583
10\t14789
11\t11932
copy values for calculator
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answer attempt 1 out of 3
model: y =
the model a good fit because

Explanation:

Step1: Calculate sums

Let \(n = 11\). Calculate \(\sum_{i = 1}^{n}x_{i}=1 + 2+\cdots+11=\frac{11\times(11 + 1)}{2}=66\), \(\sum_{i = 1}^{n}y_{i}=26340+26486+\cdots+11932 = 234835\), \(\sum_{i = 1}^{n}x_{i}^{2}=1^{2}+2^{2}+\cdots+11^{2}=\frac{11\times(11 + 1)\times(2\times11 + 1)}{6}=506\), \(\sum_{i = 1}^{n}x_{i}y_{i}=1\times26340+2\times26486+\cdots+11\times11932=1077799\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\).
Substitute the values: \(m=\frac{11\times1077799-66\times234835}{11\times506 - 66^{2}}=\frac{11855789-15499110}{5566 - 4356}=\frac{-3643321}{1210}\approx - 2994.563\).

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\).
Substitute \(m\approx - 2994.563\), \(\sum_{i = 1}^{n}x_{i}=66\), \(\sum_{i = 1}^{n}y_{i}=234835\) and \(n = 11\):
\(b=\frac{234835-(-2994.563)\times66}{11}=\frac{234835 + 197641.158}{11}=\frac{432476.158}{11}\approx39316.014\).

Step4: Write the model

The linear - regression model is \(y=-2994.563x + 39316.014\).

Step5: Analyze the residual plot (conceptually)

To determine if the model is a good fit using a residual plot, we would plot the residuals (the differences between the observed \(y\) - values and the predicted \(y\) - values from the model \(y=-2994.563x + 39316.014\)). If the points on the residual plot are randomly scattered around the horizontal axis (residual = 0), then the model is a good fit. In this case, since the sales data shows a clear decreasing trend over time, and the linear regression model is based on the least - squares method to fit a straight line to the data, we can say that the model seems to capture the general trend. If the residuals are randomly distributed around zero in the residual plot, we can conclude that the model is a good fit.

Answer:

Model: \(y=-2994.563x + 39316.014\)
The model is a good fit if the residual plot shows randomly - scattered points around the horizontal axis.