Question

type i error: a company that manufactures steel wires guarantees that the mean breaking strength (in kilonewtons) of the wires is greater than 50. they measure the strengths for a sample of wires and test $h_0: \mu = 50$ versus $h_1: \mu > 50$.
part: 0 / 3
part 1 of 3
if a type i error is made, what conclusion will be drawn regarding the mean breaking strength?
the conclusion will be that the mean breaking strength is
select
50.
part: 1 / 3
part 2 of 3
if a type ii error is made, what conclusion will be drawn regarding the mean breaking strength?
the conclusion will be that the mean breaking strength is
select
50.
part: 2 / 3
part 3 of 3
this test uses a one - tailed alternative hypothesis. explain why a one - tailed hypothesis is more appropriate than a two - tailed hypothesis in this situation.
with a one - tailed hypothesis, we can conclude that the mean breaking strength is
select

1. with a two - tailed hypothesis, we will not know whether the mean breaking strength is

select
50.

Explanation:

Part 1: Type I Error Conclusion

Step 1: Recall Type I Error Definition

A Type I error occurs when we reject the null hypothesis ($H_0: \mu = 50$) when it is actually true. The alternative hypothesis is $H_1: \mu > 50$. So, if we make a Type I error, we incorrectly conclude the alternative hypothesis is true.

Step 2: Determine the Conclusion

Since $H_1$ is $\mu > 50$, the incorrect conclusion (due to Type I error) is that the mean breaking strength is greater than 50.

Step 1: Recall Type II Error Definition

A Type II error occurs when we fail to reject the null hypothesis ($H_0: \mu = 50$) when the alternative hypothesis ($H_1: \mu > 50$) is actually true. So, we incorrectly retain the null hypothesis.

Step 2: Determine the Conclusion

Since $H_0$ is $\mu = 50$, the incorrect conclusion (due to Type II error) is that the mean breaking strength is equal to 50.

Step 1: Understand One - Tailed Hypothesis

The company wants to guarantee that the mean breaking strength is greater than 50. A one - tailed hypothesis ($H_1: \mu > 50$) focuses on detecting an effect in one direction (greater than). With a one - tailed test, if we reject $H_0$, we can conclude that the mean is greater than 50 because the alternative hypothesis is directional.

Step 2: Understand Two - Tailed Hypothesis

A two - tailed hypothesis ($H_1: \mu
eq 50$) would detect a difference in either direction (greater than or less than 50). In this case, we are only interested in whether the mean is greater than 50. If we used a two - tailed test and rejected $H_0$, we wouldn't know if the mean was greater than or less than 50, just that it was different. So a one - tailed test is more appropriate here because it aligns with the company's goal of showing the mean is greater than 50.

Step 3: Fill in the Blanks

With a one - tailed hypothesis, we can conclude that the mean breaking strength is greater than 50. With a two - tailed hypothesis, we will not know whether the mean breaking strength is greater than or less than 50.

Answer:

greater than

Part 2: Type II Error Conclusion