For the following table of​ data,
a. Draw a scatterplot.
b. Calculate the correlation coefficient.
c. Calculate the least squares line and graph it on the scatterplot.
d. Predict the​ y-value when x is 11.
x
1
2
3
4
5
6
7
8
9
10
 
y
0.4
1.0
1.4
2.4
2.9
3.4
3.4
4.4
4.9
5.4
 

Let's approach the problem step by step:

### Data:
- \( x = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
- \( y = \{0.4, 1.0, 1.4, 2.4, 2.9, 3.4, 3.4, 4.4, 4.9, 5.4\} \)

### Steps to solve:

**(a) Draw a Scatterplot:**
A scatterplot is created by plotting the points \((x, y)\) on a Cartesian plane.

**(b) Calculate the Correlation Coefficient (\(r\)):**
The correlation coefficient \(r\) measures the strength and direction of the linear relationship between two variables \(x\) and \(y\). The formula for \(r\) is:

\[
r = \frac{n\sum{xy} - \sum{x}\sum{y}}{\sqrt{(n\sum{x^2} - (\sum{x})^2)(n\sum{y^2} - (\sum{y})^2)}}
\]

where \(n\) is the number of data points.

**(c) Calculate the Least Squares Line:**
The least squares regression line is the best fit line for the data points and has the form \( y = mx + b \), where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept

The formulas to calculate the slope (\(m\)) and intercept (\(b\)) are:

\[
m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]
\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]

**(d) Predict the \( y \)-value when \( x = 11 \):**
Once we have the least squares line equation \( y = mx + b \), we can substitute \( x = 11 \) to predict \( y \).

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Let me calculate these values for you now.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you want to try calculating these values manually or use other tools, feel free to reach out if you need assistance with the process.

Would you like more details on the steps involved or any clarifications?

Here are five related questions to deepen your understanding:
1. How can you visually assess whether a linear model is appropriate for your data?
2. What does the correlation coefficient \( r \) tell you about the strength of the relationship?
3. How is the least squares method different from other methods of finding lines of best fit?
4. How do you interpret the slope and intercept in the context of a regression problem?
5. Can the correlation coefficient be negative? If yes, what does it imply?

**Tip**: When interpreting correlation, values close to 1 or -1 indicate a strong linear relationship, while values close to 0 suggest a weak or no linear relationship.

For the following table of data, a. Draw a scatterplot. b. Calculate the correlation coefficient. c. Calculate the least squares line and graph it on the scatterplot. d. Predict the y-value when x is 11.
x: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
y: 0.4, 1.0, 1.4, 2.4, 2.9, 3.4, 3.4, 4.4, 4.9, 5.4

Learn how to create a scatterplot, calculate the correlation coefficient, and find the least squares line for the given data points: x = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, y = {0.4, 1.0, 1.4, 2.4, 2.9, 3.4, 3.4, 4.4, 4.9, 5.4}. Additionally, predict the y-value when x is 11 using the regression line equation. Ideal for students in Grades 10-12.

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