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 85 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 =−4.081+0.306𝑤,
with standard error for the coefficient of 𝑤
 equal to 0.13.

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 need to perform a hypothesis test on the slope coefficient of the regression equation. 

Given:
- The regression equation is \( \hat{h} = -4.081 + 0.306w \).
- The standard error of the coefficient for \( w \) is 0.13.

### Hypothesis Testing

We are testing the null hypothesis:
- \( H_0: \beta_w = 0 \) (no relationship between wingspan and nest height)

Against the alternative hypothesis:
- \( H_1: \beta_w \neq 0 \) (there is a relationship between wingspan and nest height)

### Test Statistic
The test statistic for the slope \( \beta_w \) is calculated as:

\[
t = \frac{\hat{\beta}_w}{\text{SE}(\hat{\beta}_w)}
\]

Substituting the given values:

\[
t = \frac{0.306}{0.13} \approx 2.3538
\]

### Degrees of Freedom
The degrees of freedom (df) for this test are \( n - 2 \), where \( n \) is the sample size.

\[
df = 85 - 2 = 83
\]

### p-value Calculation
To find the p-value, we need to determine the probability of observing a t-statistic as extreme as 2.3538 (or more extreme) under the null hypothesis. We look up the two-tailed p-value for the calculated t-statistic with 83 degrees of freedom.

Using a t-distribution table or software, the p-value associated with \( t = 2.3538 \) and 83 degrees of freedom can be calculated.

Let's compute it.

The p-value for the test is approximately **0.0209**.

This p-value indicates that there is a statistically significant relationship between the wingspan of the stoach and its nest height at the 5% significance level, as the p-value is less than 0.05.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What does a p-value signify in the context of hypothesis testing?
2. How would the conclusion change if the p-value was above 0.05?
3. What assumptions are required for linear regression to be valid?
4. How do you interpret the slope of the regression line in this context?
5. How would you calculate the confidence interval for the slope coefficient?

**Tip:** When interpreting regression results, always consider both the p-value and the confidence interval for a more comprehensive understanding of the relationship.

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 85 stoaches is observed, and for each, the wingspan (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 = -4.081 + 0.306w, with standard error for the coefficient of w equal to 0.13. Determine the p-value from a test for a statistically significant linear dependence of nest height on wingspan. (Give your answer to 4 decimal places.)

Learn how to determine the p-value for testing a statistically significant linear dependence between a stoach's wingspan and its nest height. This example covers key statistical concepts such as linear regression, hypothesis testing, and p-value calculations. Suitable for undergraduate students studying introductory statistics.

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