Question

population regression function
the linear regression line shows the overall linear relationship among the plotted points. each position along the line represents where one would expect ( y ) to be given a particular ( x ), based on the linear relationship. assuming the data represents a population, the line is called a population regression function.
the population regression function in terms of the greek letter beta ( \beta ) is ( e(y) = \beta_0 + \beta_1 x ), where ( \beta_0 ) and ( \beta_1 ) are the intercept and slope parameters respectively. the notation ( e(y) ) means expected value of ( y ).
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2 2 4. population linear regression function.
the equation for the population linear regression line is
( e(y) = \beta_0 + \beta_1 x )
captions

1. if the data represents the whole population, a line that represents the relationship between and can be obtained.
2. represents the intercept parameter, which is the value of the intersection point between the line and the vertical axis.
3. represents the slope parameter, which is the ratio between the lines rise and run.
4. the equation for the population linear regression line is.

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2.2.5: linear regression line.

1. what sign does the slope of the linear regression line have if ( \beta_1 > 0 )?

• positive
• negative

1. what is the sign of ( y ) when ( x = 0 ) on the linear regression line if ( \beta_0 > 0 )?

• positive
• negative

Explanation:

1. For the slope parameter $\beta_1$, a positive value indicates that as $X$ increases, $E(Y)$ increases, which corresponds to a positive slope.
2. When $X=0$, the value of $Y$ on the regression line is $\beta_0$. If $\beta_0 > 0$, this value is positive.

Answer:

1. Positive
2. Positive