Answer:
![(14.5-13.9) -1.69 \sqrt{\frac{3.98^2}{20}+ \frac{4.03^2}{17}} = -1.63](https://tex.z-dn.net/?f=%20%2814.5-13.9%29%20-1.69%20%5Csqrt%7B%5Cfrac%7B3.98%5E2%7D%7B20%7D%2B%20%5Cfrac%7B4.03%5E2%7D%7B17%7D%7D%20%3D%20-1.63)
![(14.5-13.9) +1.69 \sqrt{\frac{3.98^2}{20}+ \frac{4.03^2}{17}} = 2.83](https://tex.z-dn.net/?f=%20%2814.5-13.9%29%20%2B1.69%20%5Csqrt%7B%5Cfrac%7B3.98%5E2%7D%7B20%7D%2B%20%5Cfrac%7B4.03%5E2%7D%7B17%7D%7D%20%3D%202.83)
And the confidence interval for this case ![-1.63 \leq \mu_1 -\mu_2 \leq 2.83](https://tex.z-dn.net/?f=%20-1.63%20%5Cleq%20%5Cmu_1%20-%5Cmu_2%20%5Cleq%202.83)
Step-by-step explanation:
We know the following info from the problem
sample mean for the group 1
the standard deviation for the group 1
the sample size for group 1
sample mean for the group 2
the standard deviation for the group 2
the sample size for group 2
We have all the conditions satisifed since we have random samples.
We want to construct a confidence interval for the true difference of means and the correct formula for this case is:
![(\bar X_1 -\bar X_2) \pm t_{\alpha/2}\sqrt{\frac{s^2_1}{n_1} +\frac{s^2_2}{n_2}}](https://tex.z-dn.net/?f=%20%28%5Cbar%20X_1%20-%5Cbar%20X_2%29%20%5Cpm%20t_%7B%5Calpha%2F2%7D%5Csqrt%7B%5Cfrac%7Bs%5E2_1%7D%7Bn_1%7D%20%2B%5Cfrac%7Bs%5E2_2%7D%7Bn_2%7D%7D)
The degrees of freedom are given :
![df = n_1 +n_2- 2 = 20+17-2=35](https://tex.z-dn.net/?f=%20df%20%3D%20n_1%20%2Bn_2-%202%20%3D%2020%2B17-2%3D35)
The confidence level is 0.9 or 90% and the significance level is
and
and the critical value for this case is:
![t_{\alpha/2} = 1.69](https://tex.z-dn.net/?f=%20t_%7B%5Calpha%2F2%7D%20%3D%201.69)
And replacing the info given we got:
![(14.5-13.9) -1.69 \sqrt{\frac{3.98^2}{20}+ \frac{4.03^2}{17}} = -1.63](https://tex.z-dn.net/?f=%20%2814.5-13.9%29%20-1.69%20%5Csqrt%7B%5Cfrac%7B3.98%5E2%7D%7B20%7D%2B%20%5Cfrac%7B4.03%5E2%7D%7B17%7D%7D%20%3D%20-1.63)
![(14.5-13.9) +1.69 \sqrt{\frac{3.98^2}{20}+ \frac{4.03^2}{17}} = 2.83](https://tex.z-dn.net/?f=%20%2814.5-13.9%29%20%2B1.69%20%5Csqrt%7B%5Cfrac%7B3.98%5E2%7D%7B20%7D%2B%20%5Cfrac%7B4.03%5E2%7D%7B17%7D%7D%20%3D%202.83)
And the confidence interval for this case ![-1.63 \leq \mu_1 -\mu_2 \leq 2.83](https://tex.z-dn.net/?f=%20-1.63%20%5Cleq%20%5Cmu_1%20-%5Cmu_2%20%5Cleq%202.83)