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 67 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=−0.815+0.544𝑤,
with standard error for the coefficient of 𝑤 equal to 0.905.

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 statistically significant linear dependence of nest height on wingspan, we first need to perform a hypothesis test on the slope of the regression line. The hypotheses for this test are:

- Null hypothesis (\( H_0 \)): The slope \( \beta_1 = 0 \) (no linear relationship between wingspan and nest height).
- Alternative hypothesis (\( H_1 \)): The slope \( \beta_1 \neq 0 \) (there is a linear relationship between wingspan and nest height).

The test statistic \( t \) for the slope is calculated using the formula:

\[
t = \frac{\text{estimated slope} - \text{hypothesized slope}}{\text{standard error of the slope}}
\]

Given:
- Estimated slope (\( \hat{\beta}_1 \)) = 0.544
- Hypothesized slope (\( \beta_1 \)) = 0 (under the null hypothesis)
- Standard error of the slope = 0.905

Substituting these values into the formula:

\[
t = \frac{0.544 - 0}{0.905} = \frac{0.544}{0.905} \approx 0.6011
\]

Next, we calculate the \( p \)-value based on this \( t \)-statistic. The degrees of freedom (\( df \)) for this test is \( n - 2 \), where \( n \) is the number of observations.

- \( n = 67 \), so \( df = 67 - 2 = 65 \).

We now use the \( t \)-distribution with 65 degrees of freedom to find the \( p \)-value for \( t \approx 0.6011 \). Since this is a two-tailed test, the \( p \)-value is the probability that the test statistic is more extreme than \( \pm t \).

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

\[
p \approx 0.5498
\]

Thus, the \( p \)-value for the test is approximately **0.5498**.

Would you like more details on this process or have any other questions?

### Related Questions:
1. How does the \( p \)-value interpretation change with different significance levels?
2. What are the assumptions for linear regression?
3. How would the results change if the sample size were different?
4. What other statistical tests can be applied to this data?
5. How can we visualize the relationship between wingspan and nest height?

### Tip:
When performing linear regression, always check if the residuals (errors) are normally distributed, as this is one of the key assumptions for the reliability of the \( p \)-value derived from the \( t \)-statistic.

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 67 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=−0.815+0.544𝑤, with a standard error for the coefficient of 𝑤 equal to 0.905. Determine the 𝑝-value from a test for a statistically significant linear dependence of nest height on wingspan.

Determine the p-value for a linear regression model that tests the relationship between wingspan and nest height in stoaches. This problem covers key statistical concepts such as hypothesis testing, t-distribution, and linear regression. Suitable for undergraduate statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation