if X=
5
6
8
10
12
13
15
16
17 and Y=
32
38
45
54
70
80
82
88
96 find regression of it

To find the regression of the given data sets \( X \) and \( Y \), we will use the method of least squares to find the equation of the best-fit line, typically expressed as:

\[
Y = a + bX
\]

where:
- \( b \) is the slope of the line.
- \( a \) is the y-intercept.

The formulas for \( a \) and \( b \) are:

\[
b = \frac{n\sum{XY} - \sum{X}\sum{Y}}{n\sum{X^2} - (\sum{X})^2}
\]

\[
a = \frac{\sum{Y} - b\sum{X}}{n}
\]

where:
- \( n \) is the number of data points.
- \( \sum{X} \) is the sum of all \( X \) values.
- \( \sum{Y} \) is the sum of all \( Y \) values.
- \( \sum{XY} \) is the sum of the product of corresponding \( X \) and \( Y \) values.
- \( \sum{X^2} \) is the sum of the squares of \( X \) values.

Let's calculate the values.The regression equation for the given data is:

\[
Y = 4.75 + 5.32X
\]

This means the best-fit line that describes the relationship between \( X \) and \( Y \) can be expressed as \( Y = 4.75 + 5.32X \).

Would you like more details or have any questions about this process?

Here are five related questions you might consider:

1. How do you interpret the slope \( b \) in the context of this data?
2. What does the y-intercept \( a \) represent in this regression equation?
3. How would you calculate the correlation coefficient for this data set?
4. How can you determine if the regression model is a good fit for the data?
5. What are the assumptions underlying linear regression?

**Tip:** In regression analysis, the slope represents the change in the dependent variable (Y) for every one-unit increase in the independent variable (X).

Learn how to perform regression analysis to find the best-fit line for given X and Y data sets. Understand the formulas for slope and y-intercept, and interpret the regression equation Y = 4.75 + 5.32X. Suitable for advanced high school or college students studying statistics and data analysis.

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