Answer :
To solve the problem of determining the value of the explanatory variable [tex]\( x \)[/tex], which gives a response variable [tex]\( y \)[/tex] value of -3.1, we need to conduct a regression analysis on the given data.
Here's a step-by-step guide:
1. Identify the Variables:
- The explanatory variable [tex]\( x \)[/tex] consists of the values: 21.2, 42.4, 46.6, 48, 69.3, 48.6, 58, 53.8, 34.4, 51.3, 42.3.
- The response variable [tex]\( y \)[/tex] consists of the values: 177.7, 119.4, 68, 34.1, -12.2, 80.7, 79.3, 97.4, 134.3, 88.5, 111.1.
2. Perform Linear Regression:
- Linear regression helps us find the best-fit line for the data. This line can be represented by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- From the analysis, we have:
- Slope ([tex]\( m \)[/tex]): -3.5201 (rounded to four decimal places)
- Intercept ([tex]\( b \)[/tex]): 254.0311 (rounded to four decimal places)
3. Use the Regression Equation:
- With the regression line equation [tex]\( y = -3.5201x + 254.0311 \)[/tex], we need to find [tex]\( x \)[/tex] when [tex]\( y = -3.1 \)[/tex].
4. Calculate Predicted Value of [tex]\( x \)[/tex]:
- Substitute [tex]\( y = -3.1 \)[/tex] in the regression equation and solve for [tex]\( x \)[/tex].
- [tex]\(-3.1 = -3.5201x + 254.0311\)[/tex]
5. Rearrange and Solve:
- First, move the intercept to the other side:
[tex]\(-3.1 - 254.0311 = -3.5201x\)[/tex]
- Simplifying the equation gives:
[tex]\(-257.1311 = -3.5201x\)[/tex]
- Divide both sides by the slope to solve for [tex]\( x \)[/tex]:
[tex]\(x = \frac{-257.1311}{-3.5201} = 73.0\)[/tex]
Therefore, the predicted explanatory value [tex]\( x \)[/tex] that would give a response value [tex]\( y \)[/tex] of -3.1 is approximately [tex]\( x = 73.0 \)[/tex] (accurate to one decimal place).
Here's a step-by-step guide:
1. Identify the Variables:
- The explanatory variable [tex]\( x \)[/tex] consists of the values: 21.2, 42.4, 46.6, 48, 69.3, 48.6, 58, 53.8, 34.4, 51.3, 42.3.
- The response variable [tex]\( y \)[/tex] consists of the values: 177.7, 119.4, 68, 34.1, -12.2, 80.7, 79.3, 97.4, 134.3, 88.5, 111.1.
2. Perform Linear Regression:
- Linear regression helps us find the best-fit line for the data. This line can be represented by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- From the analysis, we have:
- Slope ([tex]\( m \)[/tex]): -3.5201 (rounded to four decimal places)
- Intercept ([tex]\( b \)[/tex]): 254.0311 (rounded to four decimal places)
3. Use the Regression Equation:
- With the regression line equation [tex]\( y = -3.5201x + 254.0311 \)[/tex], we need to find [tex]\( x \)[/tex] when [tex]\( y = -3.1 \)[/tex].
4. Calculate Predicted Value of [tex]\( x \)[/tex]:
- Substitute [tex]\( y = -3.1 \)[/tex] in the regression equation and solve for [tex]\( x \)[/tex].
- [tex]\(-3.1 = -3.5201x + 254.0311\)[/tex]
5. Rearrange and Solve:
- First, move the intercept to the other side:
[tex]\(-3.1 - 254.0311 = -3.5201x\)[/tex]
- Simplifying the equation gives:
[tex]\(-257.1311 = -3.5201x\)[/tex]
- Divide both sides by the slope to solve for [tex]\( x \)[/tex]:
[tex]\(x = \frac{-257.1311}{-3.5201} = 73.0\)[/tex]
Therefore, the predicted explanatory value [tex]\( x \)[/tex] that would give a response value [tex]\( y \)[/tex] of -3.1 is approximately [tex]\( x = 73.0 \)[/tex] (accurate to one decimal place).
