Question #337628

The purchasing director of an industrial part factory is investigating the possibility of purchasing a new type of milling machine. He determines that the new machine will be bought if there is evidence that the parts produced have a higher mean breaking strength than those from the old machine. The population standard deviation of the breaking strength for the old machine is 10kg and for the new machine is 9 kg. A sample of 100 parts was taken from each machine. Old machine indicates a sample mean of 65kg while new machine indicates a sample mean of 72 kg. Using the 0.01 level of significance, is there evidence that the purchasing director should buy the new machine? (critical value = + 2.575 )

1
Expert's answer
2022-05-10T23:43:28-0400

The following null and alternative hypotheses need to be tested:

H0:μ1μ2H_0:\mu_1\ge\mu_2

Ha:μ1<μ2H_a:\mu_1<\mu_2

This corresponds to a left-tailed test, and a z-test for two means, with known population standard deviations will be used.

Based on the information provided, the significance level is α=0.01,\alpha = 0.01, and the critical value for a left-tailed test is zc=2.3263.z_c=-2.3263.

The rejection region for this left-tailed test is R={z:z<2.3263}.R = \{z: z <- 2.3263\}.

The z-statistic is computed as follows:


z=Xˉ1Xˉ2σ12/n1+σ22/n2z=\dfrac{\bar{X}_1-\bar{X}_2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}

=6572102/100+92/100=5.20306=\dfrac{65-72}{\sqrt{10^2/100+9^2/100}}=-5.20306

Since it is observed that z=5.20306<2.3263=zc,z = -5.20306<-2.3263= z_c,

it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=P(Z<5.20306)=0,p=P(Z<-5.20306)=0, and since p=0<0.01=α,p=0<0.01=\alpha, it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ1\mu_1

is less than μ2,\mu_2, at the α=0.01\alpha = 0.01 significance level.

Therefore, there is enough evidence to claim that the purchasing director should buy the new machine  at the α=0.01\alpha = 0.01 significance level.


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