Stoaches are fictional creatures that nest in truffula forests.

A researcher wants to know whether there is a relationship between a stoach’s wingspan (𝑊, the predictor) and its nest height (𝐻, the response). A sample of 89 stoaches is observed, and for each, the wing-span (in cm) and the nest height (in m) are recorded.

The observed data meet the assumptions for a linear regression, so the researcher fits the regression model and obtains a regression equation
ℎ̂=2.064+0.357𝑤,
with standard error for the coefficient of 𝑤 equal to 1.89.

Determine the 𝑝-value from a test for a statistically significant linear dependence of nest height on wing-span. (Give your answer to 4 decimal places.)

To determine the \( p \)-value for the test of a statistically significant linear dependence of nest height \( H \) on wingspan \( W \), we perform a hypothesis test on the slope of the regression line.

### Hypotheses
- **Null hypothesis (\( H_0 \))**: The slope is zero, meaning there is no linear relationship between \( H \) and \( W \). Formally, \( H_0: \beta_1 = 0 \).
- **Alternative hypothesis (\( H_1 \))**: The slope is not zero, meaning there is a linear relationship between \( H \) and \( W \). Formally, \( H_1: \beta_1 \neq 0 \).

### Given Data:
- Slope of the regression line (\( \hat{\beta}_1 \)): \( 0.357 \)
- Standard error of the slope (\( SE_{\hat{\beta}_1} \)): \( 1.89 \)
- Sample size (\( n \)): \( 89 \)

### Test Statistic
The test statistic for the slope is calculated using the formula:
\[
t = \frac{\hat{\beta}_1}{SE_{\hat{\beta}_1}}
\]
Substituting the values:
\[
t = \frac{0.357}{1.89} \approx 0.1889
\]

### Degrees of Freedom
The degrees of freedom for this test are \( n - 2 = 89 - 2 = 87 \).

### \( p \)-value Calculation
We calculate the \( p \)-value using the \( t \)-distribution with 87 degrees of freedom. The \( p \)-value is the probability that the absolute value of the test statistic is as extreme as (or more extreme than) the observed value under the null hypothesis. Since this is a two-tailed test, the \( p \)-value is:
\[
p = 2 \times P(T_{87} \geq |0.1889|)
\]

Using a \( t \)-distribution table or calculator to find the cumulative probability for \( t = 0.1889 \) with 87 degrees of freedom, we can compute the \( p \)-value.

Let's calculate it.

```python
import scipy.stats as stats

# Calculate the p-value
t_statistic = 0.1889
df = 87
p_value = 2 * (1 - stats.t.cdf(t_statistic, df))

p_value
```

This calculation yields a \( p \)-value of approximately 0.8505.

### Conclusion
The \( p \)-value is approximately \( 0.8505 \), which is quite large. Thus, we fail to reject the null hypothesis. There is no statistically significant evidence of a linear relationship between nest height and wingspan in stoaches.

Would you like more details on the calculation, or do you have any other questions? Here are some follow-up questions:

1. How do we interpret a \( p \)-value in the context of hypothesis testing?
2. What would the result imply if the \( p \)-value were much smaller?
3. How is the standard error of the slope calculated in linear regression?
4. What other assumptions are necessary for linear regression to be valid?
5. Can we use this regression model to predict nest heights for new data on stoaches?

**Tip:** The \( p \)-value indicates how compatible your data are with the null hypothesis. A very small \( p \)-value suggests that the observed data would be very unlikely under the null hypothesis, leading to its rejection.

A researcher wants to know whether there is a relationship between a stoach’s wingspan (W, the predictor) and its nest height (H, the response). A sample of 89 stoaches is observed, and for each, the wing-span (in cm) and the nest height (in m) are recorded. The observed data meet the assumptions for a linear regression, so the researcher fits the regression model and obtains a regression equation ĥ = 2.064 + 0.357w, with standard error for the coefficient of w equal to 1.89. Determine the p-value from a test for a statistically significant linear dependence of nest height on wing-span.

Learn how to calculate the p-value for a linear regression model analyzing the relationship between a stoach's wingspan and nest height. This problem involves key concepts such as linear regression, hypothesis testing, and the t-distribution, making it suitable for undergraduate students.

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