SOLUTION: Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna). (Reference: Hummingbirds, K. Long, W. Alther.) Suppose that in a remote part of the Grand Canyon, a random

Question 1165202: Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna). (Reference: Hummingbirds, K. Long, W. Alther.) Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.
3.7 2.9 3.8 4.2 4.8 3.1
The sample mean is x bar = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.64 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.30 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.30 grams? Use α = 0.10
Compute the z value of the sample test statistic. (Enter a number. Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Enter a number. Round your answer to four decimal places.)

Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
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.)*
RELATED QUESTIONS
Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna). (Reference:... (answered by Boreal)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.�... (answered by Boreal)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.�... (answered by ewatrrr)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.†... (answered by CPhill)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.�... (answered by Boreal)

Anna and Bill walk the same route from home to school. They don't leave at the same time. (answered by josmiceli)

Hummingbirds can fly long distances without stopping. Suppose that a hummingbird flies... (answered by rothauserc)

Hummingbirds can fly long distances without stopping. Suppose that a hummingbird flies... (answered by stanbon,MathTherapy)

Anna is half of Bills age. The sum of their ages is 36. How old are Anna and Bill (answered by greenestamps)