Question

which type of error? water pipe to be used in a certain application is required to have a mean breaking strength of more than 2000 pounds per foot. a test is made of $h_0: \mu = 2000$ versus $h_1: \mu > 2000$. there are two possible errors. identify them.
(a) the true mean is greater than 2000, but the pipe is not used.
this is a type $\boldsymbol{\
m ii}$ error.
(b) the true mean is 2000 or less, but the pipe is used.
this is a type $\boldsymbol{\
m i}$ or $\boldsymbol{\
m ii}$ (select) error.

Explanation:

for Part (a):

Step1: Recall Type I and Type II errors

Type I error: Reject \( H_0 \) when \( H_0 \) is true.
Type II error: Fail to reject \( H_0 \) when \( H_0 \) is false.

Step2: Analyze the scenario for (a)

\( H_0: \mu = 2000 \), \( H_1: \mu > 2000 \).
The true mean is greater than 2000 (so \( H_0 \) is false), but the pipe is not used (we failed to reject \( H_0 \)). This matches the definition of Type II error.

for Part (b):

Step1: Recall error definitions

Type I: Reject \( H_0 \) (use the pipe) when \( H_0 \) is true (mean ≤ 2000).
Type II: Fail to reject \( H_0 \) (don't use the pipe) when \( H_0 \) is false (mean > 2000).

Step2: Analyze the scenario for (b)

The true mean is 2000 or less (so \( H_0 \) is true), but the pipe is used (we rejected \( H_0 \)). This matches the definition of Type I error.

Part (a) Answer: Type II

Part (b) Answer: Type I

Answer:

for Part (b):

Step1: Recall error definitions

Type I: Reject \( H_0 \) (use the pipe) when \( H_0 \) is true (mean ≤ 2000).
Type II: Fail to reject \( H_0 \) (don't use the pipe) when \( H_0 \) is false (mean > 2000).

Step2: Analyze the scenario for (b)

The true mean is 2000 or less (so \( H_0 \) is true), but the pipe is used (we rejected \( H_0 \)). This matches the definition of Type I error.

Part (a) Answer: Type II

Part (b) Answer: Type I