Answer:
The 95% confidence interval for the difference between means is (-2164.21, -299.13).
The lower limit on the confidence interval is -$2164.21.
The upper limit on the confidence interval is -$299.13.
Step-by-step explanation:
The sample data is:
Gender Mean Std. dev. n
Female 2577.75 1916.29 67
Male 3809.42 2379.47 33
We have to calculate a 95% confidence interval for the difference between means, with a T-model.
The sample 1, of size n1=67 has a mean of 2577.75 and a standard deviation of 1916.29.
The sample 2, of size n2=33 has a mean of 3809.42 and a standard deviation of 2379.47.
The difference between sample means is Md=-1231.67.
![M_d=M_1-M_2=2577.75-3809.42=-1231.67](https://tex.z-dn.net/?f=M_d%3DM_1-M_2%3D2577.75-3809.42%3D-1231.67)
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{1916.29^2}{67}+\dfrac{2379.47^2}{33}}\\\\\\s_{M_d}=\sqrt{54808.468+171572.045}=\sqrt{226380.513}=475.795](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%7B1916.29%5E2%7D%7B67%7D%2B%5Cdfrac%7B2379.47%5E2%7D%7B33%7D%7D%5C%5C%5C%5C%5C%5Cs_%7BM_d%7D%3D%5Csqrt%7B54808.468%2B171572.045%7D%3D%5Csqrt%7B226380.513%7D%3D475.795)
The t-value for a 95% confidence interval is t=1.96.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
The 95% confidence interval for the difference between means is (-2164.21, -299.13).