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Run a regression analysis on the following bivariate set of data with [tex]$y$[/tex] as the response variable.

\[
\begin{array}{|r|r|}
\hline
\multicolumn{1}{|c|}{$x$} & \multicolumn{1}{c|}{$y$} \\
\hline
40.4 & 111 \\
\hline
50.6 & 51.6 \\
\hline
65.1 & -24.5 \\
\hline
50.9 & 44.1 \\
\hline
62 & 22.6 \\
\hline
50.5 & 41.3 \\
\hline
34.8 & -1.2 \\
\hline
48.5 & 45 \\
\hline
77.5 & -7.6 \\
\hline
43.7 & 50.8 \\
\hline
77.8 & -2.4 \\
\hline
66.9 & 3.4 \\
\hline
58.3 & 34.4 \\
\hline
62.3 & -20 \\
\hline
\end{array}
\]

Predict what value (on average) for the explanatory variable will give you a value of 103.2 on the response variable.

[tex]$\square$[/tex]

Answer :

To solve this problem, we need to perform a linear regression analysis on the given data set and then use the regression equation to find out which value of the explanatory variable (x) will give us a response variable (y) value of 103.2.

### Step 1: Understand the Data

We have a bivariate data set with two variables:
- [tex]\( x \)[/tex]: The explanatory variable
- [tex]\( y \)[/tex]: The response variable

The data points are as follows:

[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
40.4 & 111 \\
50.6 & 51.6 \\
65.1 & -24.5 \\
50.9 & 44.1 \\
62 & 22.6 \\
50.5 & 41.3 \\
34.8 & -1.2 \\
48.5 & 45 \\
77.5 & -7.6 \\
43.7 & 50.8 \\
77.8 & -2.4 \\
66.9 & 3.4 \\
58.3 & 34.4 \\
62.3 & -20 \\
\hline
\end{array}
\][/tex]

### Step 2: Perform Linear Regression

Linear regression involves finding the best-fit line that represents the relationship between x and y. The equation of a linear regression line is given by:

[tex]\[ y = \text{slope} \times x + \text{intercept} \][/tex]

From the regression analysis, we find:

- Slope: -1.7442
- Intercept: 123.2304

### Step 3: Use the Regression Equation

Now, we want to predict the value of x (explanatory variable) for which y (response variable) is 103.2. We substitute [tex]\( y = 103.2 \)[/tex] into the regression equation and solve for x.

[tex]\[
103.2 = -1.7442 \times x + 123.2304
\][/tex]

Rearrange the equation to solve for [tex]\( x \)[/tex]:

[tex]\[
1.7442 \times x = 123.2304 - 103.2
\][/tex]

[tex]\[
1.7442 \times x = 20.0304
\][/tex]

[tex]\[
x = \frac{20.0304}{1.7442}
\][/tex]

### Step 4: Calculate x

After performing the calculation, we find:

- Predicted x: 11.48

Thus, on average, the value of the explanatory variable [tex]\( x \)[/tex] that will give a response variable [tex]\( y \)[/tex] of 103.2 is approximately 11.48.

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