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](https://tex.z-dn.net/?f=%5Chat%20y%3D%5Calpha%20%2B%5Chat%20%5Cbeta%20x)
The (1 - <em>α</em>)% confidence interval for the regression coefficient or slope
is:
![Ci=\hat \beta \pm t_{\alpha/2, (n-2)}\times SE(\hat \beta )](https://tex.z-dn.net/?f=Ci%3D%5Chat%20%5Cbeta%20%5Cpm%20t_%7B%5Calpha%2F2%2C%20%28n-2%29%7D%5Ctimes%20SE%28%5Chat%20%5Cbeta%20%29)
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](https://tex.z-dn.net/?f=%5Chat%20Y%3D3.584756%2B0.090953%20X)
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](https://tex.z-dn.net/?f=%5Chat%20%5Cbeta%20%3D%200.090953%5C%5CSE%20%28%5Chat%20%5Cbeta%29%3D0.022162)
The critical value of <em>t</em> 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](https://tex.z-dn.net/?f=t_%7B%5Calpha%2F2%2C%20n-2%7D%3Dt_%7B0.05%2F2%2C%2010-2%7D%3Dt_%7B0.025%2C%208%7D%3D2.306)
Compute the 95% confidence interval for
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)](https://tex.z-dn.net/?f=CI%3D%5Chat%20%5Cbeta%20%5Cpm%20t_%7B%5Calpha%2F2%2C%20%28n-2%29%7D%5Ctimes%20SE%28%5Chat%20%5Cbeta%20%29%5C%5C%3D0.090953%5Cpm%202.306%5Ctimes%200.022162%5C%5C%3D0.090953%5Cpm%200.051105572%5C%5C%3D%280.039847428%2C%200.142058572%29%5C%5C%5Capprox%20%280.03985%2C%200.14206%29)
Thus, the 95% confidence interval for the true slope is (0.03985, 0.14206).