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 100 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 =−2.428+0.757𝑤,
with standard error for the coefficient of 𝑤
 equal to 0.396.

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 conduct a hypothesis test on the slope \( \beta_1 \) of the regression line.

### Step 1: State the Hypotheses
- **Null Hypothesis (\( H_0 \))**: There is no linear relationship between wingspan and nest height, i.e., \( \beta_1 = 0 \).
- **Alternative Hypothesis (\( H_1 \))**: There is a linear relationship between wingspan and nest height, i.e., \( \beta_1 \neq 0 \).

### Step 2: Calculate the Test Statistic
The test statistic for testing \( \beta_1 = 0 \) is given by:
\[
t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}
\]
where:
- \( \hat{\beta}_1 = 0.757 \) (the estimated slope)
- \( SE(\hat{\beta}_1) = 0.396 \) (the standard error of the slope)

Substituting the values:
\[
t = \frac{0.757}{0.396} \approx 1.9116
\]

### Step 3: Determine the \( p \)-value
The \( p \)-value corresponds to the probability of observing a \( t \)-value as extreme as the one calculated, under the null hypothesis. Since this is a two-tailed test, the \( p \)-value is given by:

\[
p = 2 \times P(T > |t|)
\]
where \( T \) follows a \( t \)-distribution with \( n - 2 \) degrees of freedom.

Given that the sample size \( n = 100 \), the degrees of freedom \( df = 100 - 2 = 98 \).

We can now calculate the \( p \)-value using a \( t \)-distribution table or software.

### Step 4: Calculate the \( p \)-value using software
Let me calculate this value for you.The \( p \)-value for the test is approximately \( 0.0588 \).

### Conclusion
Given that the \( p \)-value is \( 0.0588 \), which is slightly above the common significance level of \( 0.05 \), we fail to reject the null hypothesis at the 5% significance level. This suggests that there is not enough evidence to conclude a statistically significant linear dependence of nest height on wingspan in this sample.

Would you like further details or have any other questions? Here are five related questions you might find interesting:

1. How would the result change if the sample size were increased?
2. What is the interpretation of the standard error in the context of this regression?
3. How would you calculate a confidence interval for the slope?
4. What would the \( p \)-value indicate if it were below 0.05?
5. How can you test the overall fit of the regression model?

**Tip:** In regression analysis, always check the assumptions of the model, such as linearity and homoscedasticity, to ensure valid conclusions.

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 100 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 = -2.428 + 0.757w, with standard error for the coefficient of w equal to 0.396. Determine the p-value from a test for a statistically significant linear dependence of nest height on wing-span.

Learn how to determine the p-value for testing the linear dependence of nest height on wingspan in stoaches using linear regression. This problem involves key statistical concepts such as hypothesis testing, calculating test statistics, and interpreting p-values with a real-life example. Suitable for undergraduate students studying statistics.

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