User: It is known that if a signal of value μ is sent from location A, then the value received at location B is normally distributed with mean μ and standard deviation 3. That is, the random noise added to the signal is an N(0,9)
 random variable. There is reason for the people at location B to suspect that the signal value sent today is 
\mu=12. A hypothesis testing is conducted based on 6 independent signals received at location B at a significance level of 0.05 to determine if 
\mu=12 or 
\mu>12 . The average of the six signals is 13.1. What is the p-value (up to 2 non-zero digits)?
Group of answer choices

0.37

0.82

0.18

None of the above



It is known that if a signal of value μ is sent from location A, then the value received at location B is normally distributed with mean μ and standard deviation 3. That is, the random noise added to the signal is an N(0,9)
 random variable. There is reason for the people at location B to suspect that the signal value sent today is 
\mu=12. A hypothesis testing is conducted based on 6 independent signals received at location B at a significance level of 0.05 to determine if 
\mu=12 or 
\mu>12 . The average of the six signals is 13.1. What is the p-value (up to 2 non-zero digits)?
Group of answer choices

0.37

0.82

0.18

None of the above



It is known that if a signal of value μ is sent from location A, then the value received at location B is normally distributed with mean μ and standard deviation 3. That is, the random noise added to the signal is an N(0,9)
 random variable. There is reason for the people at location B to suspect that the signal value sent today is 
\mu=12. A hypothesis testing is conducted based on 6 independent signals received at location B at a significance level of 0.05 to determine if 
\mu=12 or 
\mu>12 . The average of the six signals is 13.1. What is the p-value (up to 2 non-zero digits)?
Group of answer choices

0.37

0.82

0.18

None of the above
Assistant:  Let'sLet's analyze the problem step by step.

### Given:
- Null hypothesis: \( H_0: \mu = 12 \)
- Alternative hypothesis: \( H_a: \mu > 12 \)
- Sample size: \( n = 6 \)
- Sample mean: \( \bar{x} = 13.1 \)
- Known standard deviation of each signal: \( \sigma = 3 \)
- Significance level: \( \alpha = 0.05 \)

---

### Step 1: Calculate the test statistic

Since \(\sigma\) is known, we can use a z-test for the mean:
\[
Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
where \(\mu_0 = 12\).

Calculate:
\[
Z = \frac{13.1 - 12}{3 / \sqrt{6}} = \frac{1.1}{3 / 2.449} = \frac{1.1}{1.2247} \approx 0.898
\]

---

### Step 2: Find the p-value

Since it is a **right-tailed test** (\( \mu > 12 \)), the p-value is:
\[
p = P(Z > 0.898)
\]

Use standard normal distribution table or calculator:

- \(P(Z < 0.898) \approx 0.8159\)
- So,
\[
p = 1 - 0.8159 = 0.1841 \approx 0.18
\]

---

### Step 3: Conclusion

- p-value \(\approx 0.18\)
- Compare p-value with \(\alpha = 0.05\).
- Since \(0.18 > 0.05\), we **fail to reject** \(H_0\).

---

### Final answer:

**The p-value is approximately 0.18.**

---

### Among options:
- 0.37
- 0.82
- 0.18
- None of the above

The correct choice is **0.18**.
User: https://www.youtube.com/watch?v=niWpfRyvs2U
Assistant: