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

There is sufficient evidence to conclude that the new algorithm has a lower mean completion time than the current algorithm

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{( \sigma_1 ^2 )/(n_1) + (\sigma^2_2 )/( n_2)} }`

=>
`Z = \frac{ 14.06 - 16.43 }{ \sqrt{( 3.004^2 )/(61) + (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