Question

61 sample problems. The new algorithm completes the sample problems with a mean time of 14.06 hours. The current algorithm completes the sample problems with a mean time of 16.43 hours. Assume the population standard deviation for the new algorithm is 3.004 hours, while the current algorithm has a population standard deviation of 4.568 hours. Conduct a hypothesis test at the 0.05 level of significance of the claim that the new algorithm has a lower mean completion time than the current algorithm. Le

Answer 1

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 $n_1 = n_2 = n = 61$

     The sample mean  for new algorithm is  $\= x_1 = 14.06 \ hr$

     The  standard deviation for new algorithm is  $\sigma _1 = 3.004 \ hr$

     The  sample  mean for  the current algorithm is  $\= x_2 = 16.43 \ hr$

     The standard deviation for current algorithm is  $\sigma _2 = 4.568$

     The  level of significance is $\alpha = 0.05$

The  null hypothesis is  $H_o : \mu_1 = \mu _2$

The  alternative  hypothesis is $H_a : \mu_1 < \mu_2$

Here  $\mu _1 \ and \ \mu_2$ are population mean for  new and current algorithm

Generally the test statistics is mathematically represented as

       $Z = \frac{ \= x _1 - \= x_2 }{ \sqrt{\frac{ \sigma_1 ^2 }{n_1} + \frac{\sigma^2_2 }{ n_2}} }$

=>    $Z = \frac{ 14.06 - 16.43 }{ \sqrt{\frac{ 3.004^2 }{61} + \frac{4.568^2 }{ 61}} }$

=>    $Z = -3.39$

Generally the p-value is mathematically represented as

     $p-value = P(Z < -3.39 )$

From the  z-table  

             $P(Z < -3.39 ) = 0.0003$

=>    $p-value = P(Z < -3.39 ) = 0.0003$

From the calculated value  we see that  $p-value < \alpha$  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