Answer :
Let's first understand the given power of a hypothesis test.
### Step-by-Step Solution:
1. Understanding the Power of the Test:
- The given power of the test is [tex]\( 0.4679 \)[/tex].
- The power of a test, denoted as [tex]\( 1 - \beta \)[/tex], is the probability that the test correctly rejects the null hypothesis [tex]\( H_0 \)[/tex] when the alternative hypothesis [tex]\( H_A \)[/tex] is true.
- In this case, the power is the probability of supporting the claim [tex]\( \mu < 7 \)[/tex] hours when the actual population mean is [tex]\( \mu = 5.0 \)[/tex] hours.
2. Calculating the Value of [tex]\(\beta\)[/tex]:
- Since the power of the test is [tex]\( 1 - \beta \)[/tex], we can calculate [tex]\(\beta\)[/tex] as follows:
[tex]\[ \beta = 1 - \text{Power} = 1 - 0.4679 = 0.5321 \][/tex]
- Thus, [tex]\(\beta\)[/tex] is [tex]\( 0.5321 \)[/tex].
3. Interpreting the Value of [tex]\(\beta\)[/tex]:
- The value [tex]\(\beta = 0.5321\)[/tex] represents the probability of a Type II error.
- A Type II error occurs when the test fails to reject the null hypothesis [tex]\( H_0 \)[/tex] (which is [tex]\(\mu = 7\)[/tex]) even though the alternative hypothesis [tex]\( H_A \)[/tex] ([tex]\(\mu < 7\)[/tex]) is true.
- Hence, [tex]\(\beta\)[/tex] indicates the likelihood of not recognizing the true state (i.e., we fail to conclude that [tex]\(\mu < 7\)[/tex] when in reality [tex]\(\mu = 5.0\)[/tex]).
4. Choosing the Correct Interpretation:
- We need to select the option that accurately describes the value of [tex]\(\beta = 0.5321\)[/tex] in this context.
- Option C is the correct interpretation:
[tex]\[ \text{The value } \beta = 0.5321 \text{ indicates that there is a greater than 50\% chance of failing to recognize that } \mu < 7 \text{ hours when in reality } \mu = 5.0 \text{ hours.} \][/tex]
### Final Answer:
C. The value [tex]\(\beta = 0.5321\)[/tex] indicates that there is a greater than 50% chance of failing to recognize that [tex]\(\mu < 7\)[/tex] hours when in reality [tex]\(\mu = 5.0\)[/tex] hours.
### Step-by-Step Solution:
1. Understanding the Power of the Test:
- The given power of the test is [tex]\( 0.4679 \)[/tex].
- The power of a test, denoted as [tex]\( 1 - \beta \)[/tex], is the probability that the test correctly rejects the null hypothesis [tex]\( H_0 \)[/tex] when the alternative hypothesis [tex]\( H_A \)[/tex] is true.
- In this case, the power is the probability of supporting the claim [tex]\( \mu < 7 \)[/tex] hours when the actual population mean is [tex]\( \mu = 5.0 \)[/tex] hours.
2. Calculating the Value of [tex]\(\beta\)[/tex]:
- Since the power of the test is [tex]\( 1 - \beta \)[/tex], we can calculate [tex]\(\beta\)[/tex] as follows:
[tex]\[ \beta = 1 - \text{Power} = 1 - 0.4679 = 0.5321 \][/tex]
- Thus, [tex]\(\beta\)[/tex] is [tex]\( 0.5321 \)[/tex].
3. Interpreting the Value of [tex]\(\beta\)[/tex]:
- The value [tex]\(\beta = 0.5321\)[/tex] represents the probability of a Type II error.
- A Type II error occurs when the test fails to reject the null hypothesis [tex]\( H_0 \)[/tex] (which is [tex]\(\mu = 7\)[/tex]) even though the alternative hypothesis [tex]\( H_A \)[/tex] ([tex]\(\mu < 7\)[/tex]) is true.
- Hence, [tex]\(\beta\)[/tex] indicates the likelihood of not recognizing the true state (i.e., we fail to conclude that [tex]\(\mu < 7\)[/tex] when in reality [tex]\(\mu = 5.0\)[/tex]).
4. Choosing the Correct Interpretation:
- We need to select the option that accurately describes the value of [tex]\(\beta = 0.5321\)[/tex] in this context.
- Option C is the correct interpretation:
[tex]\[ \text{The value } \beta = 0.5321 \text{ indicates that there is a greater than 50\% chance of failing to recognize that } \mu < 7 \text{ hours when in reality } \mu = 5.0 \text{ hours.} \][/tex]
### Final Answer:
C. The value [tex]\(\beta = 0.5321\)[/tex] indicates that there is a greater than 50% chance of failing to recognize that [tex]\(\mu < 7\)[/tex] hours when in reality [tex]\(\mu = 5.0\)[/tex] hours.
