To address this problem, we will perform the following steps for each part:

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(a) Least Squares Estimate of the Regression Line

The regression line is of the form $y = b_0 + b_1 x$, where:

• $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
• $b_0 = \bar{y} - b_1 \bar{x}$

Here, the given data is:

• $x = [100, 125, 125, 150, 150, 200, 200, 250, 250, 300, 300, 350, 400, 400]$
• $y = [140, 150, 180, 210, 180, 320, 270, 410, 420, 450, 400, 610, 620, 670]$

We will calculate the mean and sums required for these formulas.

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(b) Estimate of NOx Emission Rate for $x = 240$

We use the regression equation $y = b_0 + b_1 x$ to estimate $y$ for $x = 240$.

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(c) Change in NOx Emission Rate for Decreasing $x$ by 50

The change in $y$ is $\Delta y = b_1 \Delta x$, where $\Delta x = -50$.

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(d) Usefulness of Regression Line for $x = 500$

We analyze whether $x = 500$ falls within a reasonable range of the data to make predictions.

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Let me compute the necessary values for the regression equation. numpy as np

Data points

x = np.array([100, 125, 125, 150, 150, 200, 200, 250, 250, 300, 300, 350, 400, 400]) y = np.array([140, 150, 180, 210, 180, 320, 270, 410, 420, 450, 400, 610, 620, 670])

Mean of x and y

x_mean = np.mean(x) y_mean = np.mean(y)

Slope (b1) and Intercept (b0) calculations

b1 = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean) ** 2) b0 = y_mean - b1 * x_mean

Estimate for y when x = 240

x_240 = 240 y_240 = b0 + b1 * x_240

Change in y for a change in x by -50

delta_x = -50 delta_y = b1 * delta_x

Regression equation results

b0, b1, y_240, delta_y### Results:

(a) Least Squares Estimate of the Regression Line

The regression line is: $y = -53.7254 + 1.7522x$

(b) Estimated NOx Emission Rate for $x = 240$

For $x = 240$, the estimated $y$ is: $y = 366.80 \, \text{ppm}$

(c) Expected Change in NOx Emission Rate for Decreasing $x$ by 50

The expected change in $y$ when $x$ decreases by 50 is: $\Delta y = -87.61 \, \text{ppm}$

(d) Usefulness of Regression Line for $x = 500$

Using the regression line to predict for $x = 500$ is not recommended because $x = 500$ is too far outside the observed range (100–400). This would make the prediction unreliable due to potential deviations from the overall trend.

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Would you like further clarifications or additional details? Here are some follow-up questions for better understanding:

1. How is the least squares method applied to minimize errors in predictions?
2. What are the limitations of linear regression when extrapolating data?
3. Can the computed regression model be improved by using a quadratic or higher-order model?
4. How does the coefficient $b_1$ represent the rate of change in the context of this problem?
5. What are alternative methods to assess whether a model fits the data well?

Tip: Always check the residual plot to confirm that linear regression is appropriate for the data.