Answer:
1. The 95% confidence interval for the difference between means is (-5.34, 11.34).
2. The standard error of (x-bar1)-(x-bar2) is 4.
![s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{9.2195^2}{10}+\dfrac{9.4868^2}{12}}\\\\\\s_{M_d}=\sqrt{8.5+7.5}=\sqrt{16}=4](https://tex.z-dn.net/?f=s_%7BM_d%7D%3D%5Csqrt%7B%5Cdfrac%7B%5Csigma_1%5E2%7D%7Bn_1%7D%2B%5Cdfrac%7B%5Csigma_2%5E2%7D%7Bn_2%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B9.2195%5E2%7D%7B10%7D%2B%5Cdfrac%7B9.4868%5E2%7D%7B12%7D%7D%5C%5C%5C%5C%5C%5Cs_%7BM_d%7D%3D%5Csqrt%7B8.5%2B7.5%7D%3D%5Csqrt%7B16%7D%3D4)
Step-by-step explanation:
We have to calculate a 95% confidence interval for the difference between means.
The sample 1, of size n1=10 has a mean of 45 and a standard deviation of √85=9.2195.
The sample 2, of size n2=12 has a mean of 42 and a standard deviation of √90=9.4868.
The difference between sample means is Md=3.
![M_d=M_1-M_2=45-42=3](https://tex.z-dn.net/?f=M_d%3DM_1-M_2%3D45-42%3D3)
The estimated standard error of the difference between means is computed using the formula:
![s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{9.2195^2}{10}+\dfrac{9.4868^2}{12}}\\\\\\s_{M_d}=\sqrt{8.5+7.5}=\sqrt{16}=4](https://tex.z-dn.net/?f=s_%7BM_d%7D%3D%5Csqrt%7B%5Cdfrac%7B%5Csigma_1%5E2%7D%7Bn_1%7D%2B%5Cdfrac%7B%5Csigma_2%5E2%7D%7Bn_2%7D%7D%3D%5Csqrt%7B%5Cdfrac%7B9.2195%5E2%7D%7B10%7D%2B%5Cdfrac%7B9.4868%5E2%7D%7B12%7D%7D%5C%5C%5C%5C%5C%5Cs_%7BM_d%7D%3D%5Csqrt%7B8.5%2B7.5%7D%3D%5Csqrt%7B16%7D%3D4)
The critical t-value for a 95% confidence interval is t=2.086.
The margin of error (MOE) can be calculated as:
![MOE=t\cdot s_{M_d}=2.086 \cdot 4=8.34](https://tex.z-dn.net/?f=MOE%3Dt%5Ccdot%20s_%7BM_d%7D%3D2.086%20%5Ccdot%204%3D8.34)
Then, the lower and upper bounds of the confidence interval are:
![LL=M_d-t \cdot s_{M_d} = 3-8.34=-5.34\\\\UL=M_d+t \cdot s_{M_d} = 3+8.34=11.34](https://tex.z-dn.net/?f=LL%3DM_d-t%20%5Ccdot%20s_%7BM_d%7D%20%3D%203-8.34%3D-5.34%5C%5C%5C%5CUL%3DM_d%2Bt%20%5Ccdot%20s_%7BM_d%7D%20%3D%203%2B8.34%3D11.34)
The 95% confidence interval for the difference between means is (-5.34, 11.34).