Answer to Question #251702 in Statistics and Probability for Jaja

Question

Two machines are used for filling plastic bottle with a net volume of 16.0 ounces. The fill volume can be assume normal with standard deviation of q1= 0.020 and q1= 0.025 ounces. A member of quality member staff suspects that both machines fill to the same net volume, wether or not this volume is 16.0 ounces. A random sample of 10 is taken from the output of each machine. Use a=0.10

Answer 1

$n_1=10 \\ mean = \bar{x_1} = \frac{1667+16.29+...+15.88+16.99}{10} = 16.339 \\ \sigma_1 = 0.020 \\ n_2 = 10 \\ \bar{x_2} = \frac{16.19+16.49+...+16.39+16.91}{10}=16.14 \\ \sigma_2 = 0.025$

Now, if two machines are used for filling plastic bottles with a net volume of 16.0 ounces , then the net mean difference between the two machines would be zero

$H_0: \mu_1 -\mu_2 = 0 \\ H_1: \mu_1 -\mu_2 ≠ 0$

It is a two-tailed test

α=0.10

Critical value

$Z_c = 1.645$

Reject H₀ if |Z| > 1.645

Test-statistic:

$Z = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma_2^2}{n_2}}} \\ Z = \frac{16.339-16.14}{\sqrt{\frac{0.02^2}{10} + \frac{0.025^2}{10}}} \\ Z = -19.66$

Here we noticed that $|z| = 19.656 > Z_c = 1.645$

Hence, null hypothesis is rejected

Conclusion: There is enough evidence to claim that the two machines which are used for filling plastic bottles do not have a net volume of 16.0 ounces​, at the $\alpha = 0.1$ significance level