Respuesta :
Answer:
(C) The slope of the regression equation is not significantly different from zero
Step-by-step explanation:
Let's suppose that we have the following linear model:
[tex]y= \beta_o +\beta_1 X[/tex]
Where Y is the dependent variable and X the independent variable. [tex]\beta_0[/tex] represent the intercept and [tex]\beta_1[/tex] the slope.
In order to estimate the coefficients [tex]\beta_0 ,\beta_1[/tex] we can use least squares estimation.
If we are interested in analyze if we have a significant relationship between the dependent and the independent variable we can use the following system of hypothesis:
Null Hypothesis: [tex]\beta_1 = 0[/tex]
Alternative hypothesis: [tex]\beta_1 \neq 0[/tex]
Or in other wouds we want to check is our slope is significant.
In order to conduct this test we are assuming the following conditions:
a) We have linear relationship between Y and X
b) We have the same probability distribution for the variable Y with the same deviation for each value of the independent variable
c) We assume that the Y values are independent and the distribution of Y is normal
The significance level is provided and on this case is [tex]\alpha=0.05[/tex]
The standard error for the slope is given by this formula:
[tex]SE_{\beta_1}=\frac{\sqrt{\frac{\sum (y_i -\hat y_i)^2}{n-2}}}{\sqrt{\sum (X_i -\bar X)^2}}[/tex]
Th degrees of freedom for a linear regression is given by [tex]df=n-2[/tex] since we need to estimate the value for the slope and the intercept.
In order to test the hypothesis the statistic is given by:
[tex]t=\frac{\hat \beta_1}{SE_{\beta_1}}[/tex]
The confidence interval for the slope would be given by this formula:
[tex] \hat \beta_1 + t_{n-2, \alpha/2} \frac{\sqrt{\frac{\sum (y_i -\hat y_i)^2}{n-2}}}{\sqrt{\sum (X_i -\bar X)^2}}[/tex]
And using the last formula we got that the confidence interval for the slope coefficient is given by:
[tex]-0.88 < \beta_1 <1.34[/tex]
IF we analyze the confidence interval contains the value 0. So we can conclude that we don't have a significant effect of the slope on this case at 5% of significance. And the best option would be:
(C) The slope of the regression equation is not significantly different from zero
