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 44 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 =−2.238+0.247𝑤,
with standard error for the coefficient of 𝑤
 equal to 0.386.

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 testing whether there is a statistically significant linear dependence of nest height (\( H \)) on wingspan (\( W \)), we need to perform a hypothesis test for the slope of the regression line. The regression equation is:

\[
\hat{H} = -2.238 + 0.247W
\]

### Step 1: Set up the hypotheses

The hypotheses for testing the slope are:

- Null hypothesis (\( H_0 \)): The slope of the regression line is 0 (\( \beta_1 = 0 \)), which means there is no relationship between wingspan and nest height.
- Alternative hypothesis (\( H_a \)): The slope of the regression line is not 0 (\( \beta_1 \neq 0 \)), which means there is a significant relationship between wingspan and nest height.

### Step 2: Compute the test statistic

The test statistic for the slope \( \beta_1 \) is given by the formula:

\[
t = \frac{\hat{\beta}_1}{\text{SE}(\hat{\beta}_1)}
\]

Where:
- \( \hat{\beta}_1 = 0.247 \) (the estimated slope)
- \( \text{SE}(\hat{\beta}_1) = 0.386 \) (the standard error for the slope)

Substitute the values:

\[
t = \frac{0.247}{0.386}
\]

### Step 3: Calculate the test statistic

Let's calculate the value of \( t \).The test statistic \( t \) is approximately 0.6399.

### Step 4: Determine the degrees of freedom

The degrees of freedom (df) for this test is given by:

\[
\text{df} = n - 2
\]

where \( n = 44 \) is the sample size. So,

\[
\text{df} = 44 - 2 = 42
\]

### Step 5: Calculate the \( p \)-value

The \( p \)-value can be obtained by finding the two-tailed probability of observing a test statistic as extreme as \( t = 0.6399 \) with 42 degrees of freedom. We'll compute this now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can use the following method to calculate the \( p \)-value:

1. You already have the \( t \)-statistic: 0.6399.
2. Use a t-distribution table or an online calculator to find the two-tailed \( p \)-value for \( t = 0.6399 \) with 42 degrees of freedom. 
3. Multiply the single-tail probability by 2 to get the final \( p \)-value.

Would you like more details on how to compute this manually or any other questions about the analysis?

Here are 5 related questions:
1. How do you interpret a \( p \)-value in hypothesis testing?
2. What assumptions must be satisfied for linear regression to be valid?
3. How is the \( t \)-statistic related to the slope of a regression line?
4. What is the role of degrees of freedom in calculating \( p \)-values?
5. What are confidence intervals, and how are they linked to hypothesis tests?

**Tip:** Always ensure your regression model meets assumptions like linearity and homoscedasticity before relying on its 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 44 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 =−2.238+0.247w, with the standard error for the coefficient of w equal to 0.386. 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 analysis between stoach wingspan and nest height. This problem involves hypothesis testing, calculating the t-statistic, and understanding the relationship between wingspan and nest height. 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