Analysis of Simple Linear Regression

We are given the results of a simple linear regression model aimed at exploring the relationship between the length of a relationship (in years) and the percentage of correct guesses in a non-verbal communication experiment between couples.

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Question 1: What is the null hypothesis?

• Interpretation of the hypothesis options:
  • a. The slope of the regression line is zero: This would suggest that there is no linear relationship between the length of the relationship and the percentage of correct guesses.
  • b. There is a significant relation, linear or non-linear, between the two variables: This includes both linear and non-linear relationships.
  • c. There is no relation whatsoever, linear or non-linear, between the two variables: This implies no relation at all.
  • d. The slope of the regression line is significantly different from zero: This is the alternative hypothesis.
  • e. The relation between the two variables is linear: This assumes linearity but does not address statistical significance.
  • f. The slope of the regression line is either positive or negative: This just states the slope is non-zero, but doesn't specify whether it's significant.

Answer 1: a. The slope of the regression line is zero
This is the standard null hypothesis in a simple linear regression, where we test whether the slope is zero (no linear relationship).

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Question 2: From the test statistic as reported above, we can conclude that the estimated value of the slope is:

• Explanation: The test statistic for the slope is given as t = 10.6365. In linear regression, the t-statistic is calculated as the ratio of the estimated slope to its standard error.

$t = \frac{\text{Estimate of the slope}}{\text{Standard Error of the slope}}$

$t = \frac{0.7307}{0.0687} \approx 10.64$

This shows that the slope estimate is 10.64 times its standard error.

Answer 2: 10.64 times its standard error

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Question 3: From the p-value as reported above, we can conclude that the decision of the test is:

• Explanation: The reported p-value is 0 (to a very small decimal). A p-value of 0 implies that the probability of observing the test statistic by chance under the null hypothesis is extremely small. Typically, if the p-value is less than a chosen significance level (commonly 0.05), we reject the null hypothesis.

Since the p-value is effectively zero, we reject the null hypothesis.

Answer 3: Reject H0

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Conclusion of the Test:

• Since we have rejected the null hypothesis, we conclude that there is a statistically significant linear relationship between the percentage of correct guesses and the length of the relationship. Specifically, the positive slope (0.7307) suggests that as the number of years in a relationship increases, the percentage of correct guesses increases.

Conclusion:

There is a significant positive linear relationship between the length of the relationship and the percentage of correct guesses in non-verbal communication.

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Do you want further details or have any questions about this explanation?

Here are 5 related questions for further learning:

1. What are the assumptions of simple linear regression?
2. How can we interpret the intercept in this regression model?
3. What does the t-statistic tell us about the slope in linear regression?
4. Why is the p-value important in hypothesis testing?
5. How would you visualize this regression result to better understand the relationship?

Tip: Always check the assumptions of linear regression, such as linearity, independence, homoscedasticity, and normality of residuals, before interpreting results.