Question

You may need to use the appropriate technology to answer this question.

A production line operation is tested for filling weight accuracy using the following hypotheses.
Hypothesis Conclusion and Action
H0: = 16 Filling okay; keep running.
Ha: ≠ 16 Filling off standard; stop and adjust machine.
The sample size is 30 and the population standard deviation is
= 0.7.
Use
= 0.05.
(a) What would a type II error mean in this situation?
A. Rejecting H0 and stopping the process when actually no over-filling or under-filling exists.
B. Accepting H0 and stopping the process when actually no over-filling or under-filling exists.
C. Accepting H0 and letting the process continue to run when actually over-filling or under-filling exists.
D. Rejecting H0 and letting the process continue to run when actually over-filling or under-filling exists.
(b) What is the probability of making a type II error when the machine is overfilling by 0.5 ounces? (Round your answer to four decimal places.)
(c) What is the power of the statistical test when the machine is overfilling by 0.5 ounces? (Round your answer to four decimal places.)
(d) Show the power curve for this hypothesis test.
What information does it contain for the production manager?
A. The power curve shows the probability of not rejecting H0 for various possible values of . In particular it shows the probability of not stopping and adjusting the machine under a variety of underfilling and overfilling situations.
B. The power curve shows the probability of rejecting H0 for various possible values of . In particular it shows the probability of stopping and adjusting the machine under a variety of underfilling and overfilling situations.
C. The power curve shows the probability of rejecting H0 for various possible values of . In particular it shows the probability of not stopping and adjusting the machine under a variety of underfilling and overfilling situations.
D. The power curve shows the probability of not rejecting H0 for various possible values of . In particular it shows the probability of stopping and adjusting the machine under a variety of underfilling and overfilling situations.

Answer 1

A type II error would mean accepting H0 and letting the process continue to run when over-filling or under-filling actually exists. The probability of making a type II error can be calculated using the complement of the power of the statistical test. The power curve provides information about the probability of rejecting H0 under different over-filling and under-filling situations.

Step-by-step explanation:

A type II error in this situation would mean accepting H0 and letting the process continue to run when actually over-filling or under-filling exists. This means that even though there is a problem with the filling weight accuracy, the test fails to identify it and allows the production line to continue running without making necessary adjustments.

The probability of making a type II error can be calculated as the complement of the power of the statistical test. In this case, the power of the test can be calculated using the standard normal distribution and the known population standard deviation. The probability of making a type II error when the machine is overfilling by 0.5 ounces can be calculated using a similar approach.

The power curve for this hypothesis test shows the probability of rejecting H0 for various possible values of the population mean. It provides information for the production manager about the likelihood of stopping and adjusting the machine under different over-filling and under-filling situations.