College

Run a Linear Regression Analysis on the data table provided. Use this information to answer the following questions.

Note: Write each answer as an equation. Round all values to three decimal places.

1. Write the Linear Equation that models this data:
[tex]\[
\text{Linear Equation: } \square
\][/tex]

2. Use this equation to predict [tex]\( y \)[/tex] when [tex]\( x = 74.1 \)[/tex]. Enter the prediction as an equation in the form [tex]\( y = \)[/tex].
[tex]\[
\text{Prediction: } y = \square
\][/tex]

[tex]\[
\begin{array}{|r|r|}
\hline
\multicolumn{1}{|c|}{x} & \multicolumn{1}{c|}{y} \\
\hline
67.2 & 8.6 \\
\hline
69.2 & 139.5 \\
\hline
101.2 & 166.3 \\
\hline
82.8 & 138.6 \\
\hline
37.3 & 0 \\
\hline
45.2 & -29.3 \\
\hline
45 & -27.8 \\
\hline
79.9 & 157.5 \\
\hline
94.4 & 127.6 \\
\hline
87 & 114.4 \\
\hline
\end{array}
\][/tex]

Answer :

Sure! Let’s go through the process of finding the linear regression equation and using it to make predictions with the data provided.

### Step 1: Find the Linear Equation

To find the linear regression equation that models the data, we need to determine the slope and intercept of the line that best fits the given data points.

#### Data Points Provided:
- (67.2, 8.6)
- (69.2, 139.5)
- (101.2, 166.3)
- (82.8, 138.6)
- (37.3, 0)
- (45.2, -29.3)
- (45, -27.8)
- (79.9, 157.5)
- (94.4, 127.6)
- (87, 114.4)

#### Resulting Linear Equation:
The linear equation is in the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the intercept.

For the given data, the linear equation that best fits is:
[tex]\[ y = 3.191x - 146.788 \][/tex]

### Step 2: Make a Prediction

Using the linear equation, we can predict the value of [tex]\( y \)[/tex] for a given [tex]\( x \)[/tex].

#### Prediction for [tex]\( x = 74.1 \)[/tex]:
Substitute [tex]\( x = 74.1 \)[/tex] into the linear equation:
[tex]\[ y = 3.191(74.1) - 146.788 \][/tex]

When this calculation is performed, the predicted value is:
[tex]\[ y = 89.688 \][/tex]

### Final Answers:

1. Linear Equation:
[tex]\[ y = 3.191x - 146.788 \][/tex]

2. Prediction for [tex]\( x = 74.1 \)[/tex]:
[tex]\[ y = 89.688 \][/tex]

These equations provide a mathematical model of the data and a prediction based on that model.

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