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 95 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 =−2.409+0.674𝑤,
 
with standard error for the coefficient of  𝑤
  equal to 0.476.

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 a test of statistically significant linear dependence of nest height on wing-span, we need to perform a hypothesis test on the coefficient of the predictor variable \( w \) (wingspan). 

The null hypothesis \( H_0 \) is that there is no linear relationship between the nest height \( h \) and the wingspan \( w \). In terms of the coefficient \( \beta_1 \) for \( w \):

\[
H_0: \beta_1 = 0
\]
\[
H_1: \beta_1 \neq 0
\]

Given:
- The estimated coefficient for \( w \) is \( \hat{\beta_1} = 0.674 \).
- The standard error of this coefficient is \( SE(\hat{\beta_1}) = 0.476 \).

We first calculate the test statistic \( t \) using the formula:

\[
t = \frac{\hat{\beta_1}}{SE(\hat{\beta_1})}
\]

Substituting the values:

\[
t = \frac{0.674}{0.476} \approx 1.416
\]

Next, we determine the degrees of freedom (df) for the test, which is \( n - 2 \), where \( n \) is the sample size. Here, \( n = 95 \), so:

\[
df = 95 - 2 = 93
\]

We now use the \( t \)-distribution with 93 degrees of freedom to find the \( p \)-value associated with the test statistic \( t = 1.416 \). Since this is a two-tailed test, the \( p \)-value is the probability that the absolute value of a \( t \)-distribution with 93 degrees of freedom is greater than 1.416.

Using a \( t \)-distribution table or software to find the \( p \)-value:

\[
p \text{-value} \approx 0.1605
\]

Therefore, the \( p \)-value from the test is approximately **0.1605**.

Would you like more details, or do you have any questions? Here are five related questions you might consider:

1. How is the standard error of a coefficient in linear regression calculated?
2. What assumptions must be met for a linear regression model to be valid?
3. What does the \( p \)-value tell us in the context of hypothesis testing?
4. How does sample size affect the degrees of freedom and the \( p \)-value?
5. What would be the interpretation if the \( p \)-value were less than 0.05?

**Tip:** Always ensure that your data meets the assumptions of linear regression, such as linearity, independence, homoscedasticity, and normality of residuals, before trusting the model's results.

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 95 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 =−2.409+0.674W, with standard error for the coefficient of W equal to 0.476. 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 in a linear regression analysis that examines the relationship between a stoach's wingspan and nest height. This problem covers key statistical concepts such as hypothesis testing, the t-distribution, and the calculation of p-values, suitable for college or university-level students.

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