Question

Six pairs of data yield r = 0.444 and the regression equation y hat = 5x + 2. Also, y bar = 18.3. What is the best predicted value of y for x = 5?

Answer 1

Concepts and reason

The concept regression and predicting the value using the regression equation are used here.

A regression equation is a linear equation which is used to predict a dependent variable based on the independent variable. This regression equation represents the relationship between the dependent and the independent variable.

Fundamentals

The regression equation between dependent variable, $y$ and the independent variable, $x$ can be represented by the following equation:

$\hat y = a + bx$

In the above regression equation, the values $a$ and $b$ are the intercept and slope coefficient of the equation respectively.

The above regression equation can be used to predict the $y$ for future $x$ as follows:

$\hat y\left( {{\rm{pred}}} \right) = a + bx\left( {{\rm{future}}} \right)$

The “pred” represents the predicted and future represents the new values of $x$ . But the prediction of the future value will be risky outside the pattern of the observed value.

The best predicted value of $y$ for future $x$ is,

$\hat y\left( {{\rm{pred}}} \right) = \bar y$

The value of r is provided as 0.444 which indicates that there is a weak relationship between the variable x and y.

The regression equation is provided as $\hat y = 5x + 2$ and the value of $\bar y$ is 18.3.

The best predicted value of y for $x = 5$ can be computed as:

$\begin{array}{c}\\\hat y\left( {{\rm{pred}}} \right) = \bar y\\\\ = 18.3\\\end{array}$

Ans:

The best predicted value of y for $x = 5$ is 18.3.