Question

A machine in a factory is supposed to fill vials with approximately 50mL of a liquid. The quality control manager wants to test whether the machine is working properly. If the machine is found to be over or under-filling, then the assembly line will need to be shut down so that it can be re-calibrated.

a. State the Null and Alternative Hypotheses to the scenario using the correct symbols.
b. What would it mean to make a Type 1 error in this situation?
c. What would it mean to make a Type 2 error in this situation?
d. Why might it be worse to make a Type 1 error in this scenario?
e. Suppose that we fail to reject the null hypothesis. Write a conclusion statement in the con of the problem.

#2 The proportion of Americans who have frequent migraines is 15.2% according to the CDC. An acupuncturist claims that her treatment can reduce this figure significantly. A random sample of 675 Americans is administered the acupuncturist's treatment and 101 report experiencing migraines.

a. State hypotheses to the scenario using the correct symbols.
b. What is the sample proportion? (Round to 2 decimal places)
c. Suppose the P-value is calculated to be 0.0276. What would your decision be for this test using α = 0.05? Would you:

Accept the null,
Reject the null, or
Fail to reject the null

Answer 1

The null hypothesis is that the machine fills vials with 50mL, while the alternative hypothesis is that it does not. A Type 1 error means mistakenly shutting down the assembly line when the machine is accurate, and a Type 2 error means overlooking an actual error in the filling process.

Step-by-step explanation:

To address the scenario involving the vials being filled with liquid, we will construct hypotheses to determine if the machine is functioning properly. The null hypothesis (H0) for this situation would be that the mean volume of liquid in the vials is 50mL (H0: μ = 50). The alternative hypothesis (Ha) would be that the mean volume is not 50mL (Ha: μ ≠ 50).

A Type 1 error would occur if we incorrectly conclude that the machine is not filling the vials properly (reject H0) when, in fact, it is functioning correctly. A Type 2 error would happen if we fail to detect that the machine is malfunctioning (we do not reject H0) when it is actually over or under-filling.

It might be considered worse to make a Type 1 error in this scenario because stopping the assembly line for recalibration is costly and unnecessary if the machine is actually working properly. If we fail to reject the null hypothesis, our conclusion might state, "Based on our quality control tests, there is insufficient evidence to claim that the filling machine is not dispensing the correct volume of liquid."