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 greater than 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.
equal to
not equal to
less than
greater than
less than or greater than

Explanation:

Part 1: Type I Error

Step1: Recall Type I Error Definition

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

Step2: Match with the Question

The question asks for the conclusion when a Type I error is made. From the dropdown, the option "greater than" (for the relation to 50) is correct as we falsely conclude $\mu > 50$.

Part 2: Type II Error

Step1: Recall Type II Error Definition

A Type II error occurs when we fail to reject the null hypothesis ($H_0$) when it is actually false. So, we should have rejected $H_0$ (since $H_1: \mu > 50$ is true), but we don't. So we conclude that $H_0$ is true, i.e., $\mu = 50$ (or fail to conclude $\mu > 50$).

Step2: Match with the Question

The question asks for the conclusion when a Type II error is made. From the dropdown, the option "equal to" (since we fail to reject $H_0: \mu = 50$) is correct.

Part 1 Answer: greater than 50 (the option "greater than" in the dropdown for Part 1)

Part 2 Answer: equal to 50 (the option "equal to" in the dropdown for Part 2)

(Note: Since the problem has two sub - parts related to hypothesis testing errors, the discipline is Mathematics (subfield: Statistics) and the solution uses the Step - by - Step format as it involves logical deductions about hypothesis testing concepts.)

Answer:

Step1: Recall Type II Error Definition

A Type II error occurs when we fail to reject the null hypothesis ($H_0$) when it is actually false. So, we should have rejected $H_0$ (since $H_1: \mu > 50$ is true), but we don't. So we conclude that $H_0$ is true, i.e., $\mu = 50$ (or fail to conclude $\mu > 50$).

Step2: Match with the Question

The question asks for the conclusion when a Type II error is made. From the dropdown, the option "equal to" (since we fail to reject $H_0: \mu = 50$) is correct.

Part 1 Answer: greater than 50 (the option "greater than" in the dropdown for Part 1)

Part 2 Answer: equal to 50 (the option "equal to" in the dropdown for Part 2)

(Note: Since the problem has two sub - parts related to hypothesis testing errors, the discipline is Mathematics (subfield: Statistics) and the solution uses the Step - by - Step format as it involves logical deductions about hypothesis testing concepts.)