Question

A Pharmaceutical Company uses a machine to pour cold medicine into bottles in such a way that the s. d. of the weights is 0.15 oz. A new machine is tested on 68 bottles and the s. d. for this sample is 0.12 oz. The Dayton Machine company, which manufactures the new machine claims that it fills bottle with a lower variation. At a 1% level of significance test the claim made by the Dayton Machine Company. For all test of hypothesis problems,
1. Set up the null and alternate hypothesis
2. Calculate the test statistic.
3. find the critical value(s) or P value.
4. Make a decision.
5. Write the conclusion in terms of the problem.

Answer 1

Answer:

1)  H0 :  σ₁ ²≥ σ₂²;   Ha:  σ₁² < σ₂²

2)  χ²= 43.52

3) The critical region is χ²≤ 5.23

4) Reject the alternate hypothesis.

5) We conclude that the alternate hypothesis is false and accept the null hypothesis.

Step-by-step explanation:

The claim is that it fills bottle with a lower variation which is the alternate hypothesis

1)  Ha:  σ₁² < σ₂² where  σ₁² is the variation of the new machine and σ₂² is the variation of the old machine.

The null hypothesis is opposite of alternate hypothesis H0 :  σ₁ ²≥ σ₂²

2) The test statistic is χ²= ns²/σ ² which under H0 has  χ² distribution with n-1 degrees of freedom assuming the population is normal.

The calculated χ²= ns²/σ ² = 68( 0.12)²/ (0.15)²=0.9792/0.0225= 43.52

3) The critical region is entirely in the left tail. χ²≤χ²( 0.99)(15)= 5.23

4) The alternate hypothesis is false hence reject it.

5) The calculated χ²=  43.52 does not lie in the critical region χ²≤ 5.23 therefore H0 is accepted and concluded that new machine does not fill bottles with a lower variation.