Question

Ten college students were randomly selected. Their grade point averages​ (GPAs) when they entered the program were between 3.5 and 4.0. The​ students' GPAs on entering the program​ (x) and their current GPAs​ (y) were recorded. The regression analysis is given below. Use the analysis to find a​ 95% confidence interval for the true slope.
The regression equation is y=3.584756+0.090953x with R-sq=0.001849 and 10-2=8 degrees of freedom

The regression equation is

ModifyingAbove y with caret

equals3.584756plus

​0.090953x,

with

​R-sqequals

​0.001849,

and

10minus

2equals

8

degrees of freedom.

Predictor

Coeff

​SE(Coeff)

T

P

Constant

3.584756

0.078183

45.85075

5.66x10-11

Entering GPA

0.090953

0.022162

4.103932

0.003419

Answer 1

Answer:

The 95% confidence interval for the true slope is (0.03985, 0.14206).

Step-by-step explanation:

For the regression equation:

$\hat y=\alpha +\hat \beta x$

The (1 - α)% confidence interval for the regression coefficient or slope $(\hat \beta )$ is:

$Ci=\hat \beta \pm t_{\alpha/2, (n-2)}\times SE(\hat \beta )$

The regression equation for current GPA (Y) of students based on their GPA's when entering the program (X) is:

$\hat Y=3.584756+0.090953 X$

The summary of the regression analysis is:

Predictor          Coefficient             SE             t-stat            p-value

Constant             3.584756          0.078183       45.85075      5.66 x 10⁻¹¹

Entering GPA   0.090953          0.022162        4.103932       0.003419

The regression coefficient and standard error are:

$\hat \beta = 0.090953\\SE (\hat \beta)=0.022162$

The critical value of t  for 95% confidence level and 8 degrees of freedom is:

$t_{\alpha/2, n-2}=t_{0.05/2, 10-2}=t_{0.025, 8}=2.306$

Compute the 95% confidence interval for $(\hat \beta )$ as follows:

$CI=\hat \beta \pm t_{\alpha/2, (n-2)}\times SE(\hat \beta )\\=0.090953\pm 2.306\times 0.022162\\=0.090953\pm 0.051105572\\=(0.039847428, 0.142058572)\\\approx (0.03985, 0.14206)$

Thus, the 95% confidence interval for the true slope is (0.03985, 0.14206).