Question 1165201
The user is asking for the calculation of the $Z$-test statistic and the corresponding $P$-value for a one-sample hypothesis test concerning a population mean ($\mu$), where the population standard deviation ($\sigma$) is known.

### Given Information:

  * **Population Mean (Hypothesized), $\mu_0$:** $4.9\%$
  * **Population Standard Deviation (Known), $\sigma$:** $2.6\%$
  * **Sample Size, $n$:** $10$
  * **Sample Mean, $\bar{x}$:** $5.38\%$
  * **Level of Significance, $\alpha$:** $0.01$

### Hypothesis Formulation:

  * **Null Hypothesis ($H_0$):** The mean dividend yield is equal to $4.9\%$ ($\mu = 4.9$).
  * **Alternative Hypothesis ($H_1$):** The mean dividend yield is higher than $4.9\%$ ($\mu > 4.9$). (This is a right-tailed test.)

### (a) Compute the $Z$ value of the sample test statistic.

The formula for the $Z$-test statistic for a sample mean when $\sigma$ is known is:
$$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$

### (c) Find the $P$-value.

Since this is a right-tailed test, the $P$-value is the probability of observing a $Z$-score as extreme as the calculated test statistic or more extreme (i.e., further to the right).
$$P\text{-value} = P(Z \ge Z_{\text{test}}) = 1 - P(Z < Z_{\text{test}})$$

I will use the code interpreter to perform these calculations. I will use the provided $Z$-value of $0.64$ to confirm the calculation if possible, and then use the calculated $Z$-value to find the $P$-value. Since the user provided the $Z$-value as "0.640" I will use my calculated value but confirm if it matches the provided one.

```python?code_reference&code_event_index=2
import numpy as np
from scipy.stats import norm

# Given parameters
mu_0 = 4.9  # Hypothesized population mean (%)
sigma = 2.6 # Population standard deviation (%)
n = 10      # Sample size
x_bar = 5.38 # Sample mean (%)

# --- Part (a): Compute the Z value of the sample test statistic ---
# 1. Calculate the standard error of the mean
SE = sigma / np.sqrt(n)

# 2. Calculate the Z-score
Z_test = (x_bar - mu_0) / SE

# --- Part (c): Find the P-value ---
# Since H1 is mu > 4.9 (right-tailed test), P-value = 1 - P(Z < Z_test)
P_value = 1 - norm.cdf(Z_test)

# Output results rounded as requested
Z_rounded = round(Z_test, 2)
P_rounded = round(P_value, 4)

print(f"Standard Error (SE): {SE}")
print(f"Z-test statistic: {Z_test}")
print(f"Z-test statistic (rounded to 2 d.p.): {Z_rounded}")
print(f"P-value: {P_value}")
print(f"P-value (rounded to 4 d.p.): {P_rounded}")

```

```text?code_stdout&code_event_index=2
Standard Error (SE): 0.8221921916437787
Z-test statistic: 0.5838051064926233
Z-test statistic (rounded to 2 d.p.): 0.58
P-value: 0.27967572112728867
P-value (rounded to 4 d.p.): 0.2797

```

The problem asks for the $Z$-test statistic and the $P$-value for a one-tailed hypothesis test.

### 1\. Compute the $Z$ value of the sample test statistic.

The formula for the $Z$-test statistic is:
$$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$

Given values: $\bar{x}=5.38$, $\mu_0=4.9$, $\sigma=2.6$, $n=10$.

1.  **Calculate the Standard Error (SE):**
    $$\text{SE} = \frac{2.6}{\sqrt{10}} \approx 0.82219$$

2.  **Calculate the $Z$-score:**
    $$Z = \frac{5.38 - 4.9}{0.82219} = \frac{0.48}{0.82219}$$
    $$Z \approx 0.5838$$

Rounding the answer to two decimal places:
$$\mathbf{Z \approx 0.58}$$

***Note:*** *The provided $Z$ value of $0.640$ is slightly different from the calculated value of $0.58$. We will use the mathematically derived value of $0.58$ for consistency.*

### 2\. Find the $P$-value.

Since the alternative hypothesis is $H_1: \mu > 4.9$ (a right-tailed test), the $P$-value is the area under the standard normal curve to the right of the test statistic $Z = 0.5838$.

$$P\text{-value} = P(Z \ge 0.5838) = 1 - P(Z < 0.5838)$$
$$P\text{-value} \approx 1 - 0.720324$$
$$P\text{-value} \approx 0.279676$$

Rounding the answer to four decimal places:
$$\mathbf{P\text{-value} \approx 0.2797}$$