Answer:
The decision rule is
Reject the null hypothesis
The test statistics is ![t = -2.699](https://tex.z-dn.net/?f=t%20%3D%20-2.699)
Step-by-step explanation:
From the question we are told that
The first sample size is ![n_1 = 15](https://tex.z-dn.net/?f=n_1%20%3D%2015)
The mean at first deployment is ![\= x _1 = \$ 150 000](https://tex.z-dn.net/?f=%5C%3D%20x%20_1%20%3D%20%20%5C%24%20150%20000)
The standard deviation is
The second sample size is ![n_2 = 25](https://tex.z-dn.net/?f=n_2%20%3D%2025)
The mean at second deployment is ![\= x_2 = \$ 180000](https://tex.z-dn.net/?f=%5C%3D%20x_2%20%3D%20%5C%24%20180000)
The standard deviation is ![s_2 = \$ 30 000](https://tex.z-dn.net/?f=s_2%20%3D%20%5C%24%2030%20000)
The null hypothesis is ![H_o : \mu_1 = \mu_2](https://tex.z-dn.net/?f=H_o%20%3A%20%5Cmu_1%20%3D%20%5Cmu_2)
The alternative hypothesis is ![H_a : \mu_1 \ne \mu_2](https://tex.z-dn.net/?f=H_a%20%3A%20%5Cmu_1%20%5Cne%20%20%5Cmu_2)
The level of significance is ![\alpha = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.05)
Generally the degree of freedom is mathematically represented as
![df =n_1 + n_2 -2](https://tex.z-dn.net/?f=df%20%3Dn_1%20%2B%20n_2%20-2)
=> ![df =40 -2](https://tex.z-dn.net/?f=df%20%3D40%20-2)
=> ![df =38](https://tex.z-dn.net/?f=df%20%3D38)
Generally the pooled variance is mathematically represented as
![s_p^2 = \frac{x(n_1 -1) s_1^2 + (n_2 - 1)s_2^2}{df}](https://tex.z-dn.net/?f=s_p%5E2%20%20%3D%20%20%5Cfrac%7Bx%28n_1%20-1%29%20s_1%5E2%20%2B%20%28n_2%20-%201%29s_2%5E2%7D%7Bdf%7D)
=> ![s_p^2 = \frac{(15 -1) 40000^2 + (25 - 1)30000^2}{38}](https://tex.z-dn.net/?f=s_p%5E2%20%20%3D%20%20%5Cfrac%7B%2815%20-1%29%2040000%5E2%20%2B%20%2825%20-%201%2930000%5E2%7D%7B38%7D)
=> ![s_p^2 = 1.1579 * 10^{9}](https://tex.z-dn.net/?f=s_p%5E2%20%20%3D%201.1579%20%2A%2010%5E%7B9%7D)
Generally the test statistics is mathematically represented as
![t = \frac{ \= x_1 - \= x_2 }{\sqrt{s_p [\frac{1}{n_1} + \frac{1}{n_2} ]} }](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%20%5C%3D%20x_1%20-%20%5C%3D%20x_2%20%7D%7B%5Csqrt%7Bs_p%20%5B%5Cfrac%7B1%7D%7Bn_1%7D%20%2B%20%5Cfrac%7B1%7D%7Bn_2%7D%20%5D%7D%20%7D)
=> ![t = \frac{ 150000 - 180000 }{\sqrt{1.11579 *10^{9} [\frac{1}{15} + \frac{1}{25} ]} }](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%20150000%20-%20180000%20%7D%7B%5Csqrt%7B1.11579%20%2A10%5E%7B9%7D%20%5B%5Cfrac%7B1%7D%7B15%7D%20%2B%20%5Cfrac%7B1%7D%7B25%7D%20%5D%7D%20%7D)
=> ![t = -2.699](https://tex.z-dn.net/?f=t%20%3D%20-2.699)
Generally from the t distribution table the probability corresponding to the t statistics value to the left is
![t_{-2.699 , 38} = 0.00516046](https://tex.z-dn.net/?f=t_%7B-2.699%20%2C%2038%7D%20%3D%200.00516046)
Generally the p -value is mathematically represented as
![p-value = 2* t_{-2.699, 38}](https://tex.z-dn.net/?f=p-value%20%3D%202%2A%20%20t_%7B-2.699%2C%2038%7D)
=> ![p-value = 2* 0.00516046](https://tex.z-dn.net/?f=p-value%20%3D%202%2A%20%200.00516046)
=> ![p-value = 0.01032](https://tex.z-dn.net/?f=p-value%20%3D%200.01032)
From the obtained value we see that the
hence
The decision rule is
Reject the null hypothesis