x values are 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19

y vaues 51.94, 52.38,48.12,45.46,44.8,42.04,38.48,35.12,32.36,32.5,26.34,26.98,25.02,21.66,17.4

Thank you for the clarification! I will now calculate the regression line for the updated \(x\) and \(y\) values.

The data provided is:

\[
\begin{array}{|c|c|}
x & y \\
\hline
5  & 51.94 \\
6  & 52.38 \\
7  & 48.12 \\
8  & 45.46 \\
9  & 44.80 \\
10 & 42.04 \\
11 & 38.48 \\
12 & 35.12 \\
13 & 32.36 \\
14 & 32.50 \\
15 & 26.34 \\
16 & 26.98 \\
17 & 25.02 \\
18 & 21.66 \\
19 & 17.40 \\
\end{array}
\]

I will compute the regression line now.# Updated data as per user input
x_new = np.array([5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19])
y_new = np.array([51.94, 52.38, 48.12, 45.46, 44.80, 42.04, 38.48, 35.12, 32.36, 
                  32.50, 26.34, 26.98, 25.02, 21.66, 17.40])

# Perform linear regression on the new data
A_new = np.vstack([x_new, np.ones(len(x_new))]).T
m_new, b_new = np.linalg.lstsq(A_new, y_new, rcond=None)[0]

m_new, b_new  # new slope and interceptThe updated equation of the regression line is:

\[
y = -2.49x + 65.87
\]

Would you like further explanation on the steps, or have any other questions?

Here are 5 additional questions to consider:
1. How can this regression line be used to predict \(y\) for a new \(x\) value?
2. What does the slope of \(-2.49\) indicate about the relationship between \(x\) and \(y\)?
3. How would you assess the accuracy of this regression model?
4. Can you calculate the residuals for each data point to check model fit?
5. What effect would removing extreme data points have on the regression line?

**Tip:** The slope in a linear regression line tells you the rate of change. In this case, for each unit increase in \(x\), \(y\) decreases by approximately 2.49.

Explore the linear regression analysis for the given x and y values: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 51.94, 52.38, 48.12, 45.46, 44.80, 42.04, 38.48, 35.12, 32.36, 32.50, 26.34, 26.98, 25.02, 21.66, 17.40 respectively. Understand the slope, intercept, and application of the regression line.

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