Answer:
There is sufficient evidence to conclude that the new algorithm has a lower mean completion time than the current algorithm
Step-by-step explanation:
From the question we are told that
The sample size for each algorithm is ![n_1 = n_2 = n = 61](https://tex.z-dn.net/?f=n_1%20%3D%20%20n_2%20%20%3D%20%20n%20%20%3D%20%2061)
The sample mean for new algorithm is ![\= x_1 = 14.06 \ hr](https://tex.z-dn.net/?f=%5C%3D%20x_1%20%3D%20%2014.06%20%5C%20%20hr)
The standard deviation for new algorithm is ![\sigma _1 = 3.004 \ hr](https://tex.z-dn.net/?f=%5Csigma%20_1%20%20%3D%20%203.004%20%5C%20hr)
The sample mean for the current algorithm is ![\= x_2 = 16.43 \ hr](https://tex.z-dn.net/?f=%5C%3D%20x_2%20%3D%20%2016.43%20%5C%20hr)
The standard deviation for current algorithm is ![\sigma _2 = 4.568](https://tex.z-dn.net/?f=%5Csigma%20%20_2%20%20%3D%20%204.568)
The level of significance is ![\alpha = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.05)
The null hypothesis is ![H_o : \mu_1 = \mu _2](https://tex.z-dn.net/?f=H_o%20%20%3A%20%20%5Cmu_1%20%3D%20%20%5Cmu%20_2)
The alternative hypothesis is ![H_a : \mu_1 < \mu_2](https://tex.z-dn.net/?f=H_a%20%20%20%3A%20%20%5Cmu_1%20%3C%20%20%5Cmu_2)
Here
are population mean for new and current algorithm
Generally the test statistics is mathematically represented as
![Z = \frac{ \= x _1 - \= x_2 }{ \sqrt{\frac{ \sigma_1 ^2 }{n_1} + \frac{\sigma^2_2 }{ n_2}} }](https://tex.z-dn.net/?f=Z%20%3D%20%20%5Cfrac%7B%20%5C%3D%20x%20_1%20-%20%20%5C%3D%20x_2%20%7D%7B%20%5Csqrt%7B%5Cfrac%7B%20%5Csigma_1%20%5E2%20%20%7D%7Bn_1%7D%20%2B%20%5Cfrac%7B%5Csigma%5E2_2%20%7D%7B%20n_2%7D%7D%20%20%7D)
=> ![Z = \frac{ 14.06 - 16.43 }{ \sqrt{\frac{ 3.004^2 }{61} + \frac{4.568^2 }{ 61}} }](https://tex.z-dn.net/?f=Z%20%3D%20%20%5Cfrac%7B%2014.06%20-%2016.43%20%7D%7B%20%5Csqrt%7B%5Cfrac%7B%20%203.004%5E2%20%20%7D%7B61%7D%20%2B%20%5Cfrac%7B4.568%5E2%20%7D%7B%2061%7D%7D%20%20%7D)
=> ![Z = -3.39](https://tex.z-dn.net/?f=Z%20%3D%20-3.39)
Generally the p-value is mathematically represented as
![p-value = P(Z < -3.39 )](https://tex.z-dn.net/?f=p-value%20%20%3D%20%20P%28Z%20%20%3C%20%20-3.39%20%29)
From the z-table
![P(Z < -3.39 ) = 0.0003](https://tex.z-dn.net/?f=P%28Z%20%20%3C%20%20-3.39%20%29%20%20%3D%20%200.0003)
=> ![p-value = P(Z < -3.39 ) = 0.0003](https://tex.z-dn.net/?f=p-value%20%20%3D%20%20P%28Z%20%20%3C%20%20-3.39%20%29%20%20%3D%20%200.0003)
From the calculated value we see that
hence the null hypothesis is rejected
Hence we can conclude that there is sufficient evidence to conclude that the new algorithm has a lower mean completion time than the current algorithm