Question 1165202
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), $\mu_0$:** $4.30$ grams
  * **Population Standard Deviation (Known), $\sigma$:** $0.64$ grams
  * **Sample Size, $n$:** $6$
  * **Sample Mean, $\bar{x}$:** $3.75$ grams
  * **Level of Significance, $\alpha$:** $0.10$

### Hypothesis Formulation:

  * **Null Hypothesis ($H_0$):** The mean weight is equal to $4.30$ grams ($\mu = 4.30$).
  * **Alternative Hypothesis ($H_1$):** The mean weight is less than $4.30$ grams ($\mu < 4.30$). (This is a left-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 left-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 left).
$$P\text{-value} = P(Z \le Z_{\text{test}})$$

I will use the code interpreter to perform these calculations.

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

# Given parameters
mu_0 = 4.30  # Hypothesized population mean
sigma = 0.64 # Population standard deviation
n = 6        # Sample size
x_bar = 3.75 # 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 < 4.30 (left-tailed test), P-value = P(Z <= Z_test)
P_value = 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): 0.26127890589687236
Z-test statistic: -2.105030247704293
Z-test statistic (rounded to 2 d.p.): -2.11
P-value: 0.017644337582746063
P-value (rounded to 4 d.p.): 0.0176

```

Here are the required calculations for the hypothesis test:

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

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

Using the given values ($\bar{x}=3.75$, $\mu_0=4.30$, $\sigma=0.64$, $n=6$):

1.  **Calculate the Standard Error (SE):**
    $$\text{SE} = \frac{0.64}{\sqrt{6}} \approx 0.26128$$

2.  **Calculate the $Z$-score:**
    $$Z = \frac{3.75 - 4.30}{0.26128} = \frac{-0.55}{0.26128}$$
    $$Z \approx -2.1050$$

Rounding the answer to two decimal places:
$$\mathbf{Z \approx -2.11}$$

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

Since the alternative hypothesis is $H_1: \mu < 4.30$ (a left-tailed test), the $P$-value is the area under the standard normal curve to the left of the test statistic $Z = -2.1050$.

$$P\text{-value} = P(Z \le -2.1050)$$
$$P\text{-value} \approx 0.017644$$

Rounding the answer to four decimal places:
$$\mathbf{P\text{-value} \approx 0.0176}$$

-----

*(For context: Since the $P$-value $(0.0176)$ is less than the significance level $\alpha$ $(0.10)$, we would reject the null hypothesis and conclude that the data indicates the mean weight of these birds in this part of the Grand Canyon is indeed less than $4.30$ grams.)*