The computer calculated the following linear regression equation: f=38.03+0.07*t 
   (in which a0,a1 are averages derived from the 55 samples) and also the following other information points: Sa0=14.2, Sa1=0.05, r^2=0.41

Your general task is to verify if there is a correlation or not, using a hypothesis test at 1% significance.

Interpret the estimated slope coefficient! What is the null hypothesis and what is the alternative hypothesis? Which test do you have to conduct and why?

### 1. **Interpreting the Estimated Slope Coefficient**

The regression equation is given as:
\[ f = 38.03 + 0.07 \cdot t \]
The slope coefficient \( a_1 = 0.07 \) means that for each unit increase in \( t \), the expected value of \( f \) increases by 0.07 units on average. 

### 2. **Null and Alternative Hypothesis**

To test for correlation (i.e., to determine whether the slope \( a_1 \) is significantly different from zero), the hypotheses are:

- **Null Hypothesis \( H_0 \):** \( a_1 = 0 \) (No correlation, meaning \( t \) does not affect \( f \))
- **Alternative Hypothesis \( H_1 \):** \( a_1 \neq 0 \) (There is a correlation, meaning \( t \) affects \( f \))

### 3. **Test to Conduct and Why**

We need to perform a **t-test** on the slope coefficient \( a_1 \). This is because we want to determine if the slope is significantly different from zero, which directly tests for the presence of a linear relationship.

### 4. **Test Statistic for the Slope \( a_1 \)**

The test statistic for a slope \( a_1 \) in linear regression is given by:
\[
t = \frac{a_1}{S_{a_1}}
\]
Where:
- \( a_1 = 0.07 \) (estimated slope coefficient)
- \( S_{a_1} = 0.05 \) (standard error of the slope)

Let's calculate the \( t \)-statistic:

\[
t = \frac{0.07}{0.05} = 1.4
\]

### 5. **Critical Value for \( t \)-distribution**

With 55 samples, the degrees of freedom \( df = 55 - 2 = 53 \). For a two-tailed test at a 1% significance level (\( \alpha = 0.01 \)), we need to compare the calculated \( t \)-statistic against the critical value of \( t \) for 53 degrees of freedom.

Using a \( t \)-table, the critical \( t \)-value for a two-tailed test with \( df = 53 \) at \( \alpha = 0.01 \) is approximately \( t_{crit} \approx 2.67 \).

### 6. **Decision: Reject or Fail to Reject \( H_0 \)?**

Since \( |t| = 1.4 \) is less than \( t_{crit} = 2.67 \), we **fail to reject the null hypothesis** at the 1% significance level. 

This means that there is **insufficient evidence to conclude** that there is a statistically significant correlation between \( t \) and \( f \) at the 1% level.

### Conclusion:
- **Interpretation of the slope:** The slope \( 0.07 \) indicates a small positive relationship between \( t \) and \( f \), but this relationship is not statistically significant at the 1% level.
- **Correlation conclusion:** There is no statistically significant correlation between \( t \) and \( f \) at the 1% significance level.

---

Would you like more details on any part of this process? 

Here are 5 related questions you might find interesting:
1. How does increasing the sample size affect the hypothesis test results?
2. What is the interpretation of the \( r^2 \) value of 0.41 in this context?
3. How would the decision change if the significance level were 5% instead of 1%?
4. What does the value \( S_{a_0} = 14.2 \) tell us about the intercept?
5. What are the assumptions of linear regression that need to be satisfied for this test to be valid?

**Tip:** Always check the \( r^2 \) value to understand how much of the variation in the dependent variable is explained by the independent variable in regression models.

The computer calculated the following linear regression equation: f=38.03+0.07*t (in which a0,a1 are averages derived from the 55 samples) and also the following other information points: Sa0=14.2, Sa1=0.05, r^2=0.41. Your general task is to verify if there is a correlation or not, using a hypothesis test at 1% significance. Interpret the estimated slope coefficient! What is the null hypothesis and what is the alternative hypothesis? Which test do you have to conduct and why?

Learn how to verify the correlation in a linear regression model using a t-test. With 55 samples, the slope coefficient is 0.07, and at a 1% significance level, this guide explains how to test if there is a significant correlation between the variables, including the interpretation of the slope, null and alternative hypotheses, and decision criteria. Suitable for undergraduate-level statistics or economics 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