Question

How do we determine whether the actual slope of the regression line is not zero?

Answer 1

For this we used testing of hypothesis

State the Hypotheses

If there is a significant linear relationship between the independent variable X and the dependent variable Y, the slope will not equal zero.

Ho: Β1 = 0

Ha: Β1 ≠ 0

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.

• Significance level. Often, researchers choosesignificance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use a linear regression t-test (described in the next section) to determine whether the slope of the regression line differs significantly from zero.

Analyze Sample Data

Using sample data, find the standard error of the slope, the slope of the regression line, the degrees of freedom, the test statistic, and the P-value associated with the test statistic.

SE = sb1 = sqrt [ Σ(yi - ŷi)2 / (n - 2) ] / sqrt [ Σ(xi - x)

• Degrees of freedom. For simple linear regression (one independent and one dependent variable), thedegrees of freedom (DF) is equal to:

  DF = n - 2

  where n is the number of observations in the sample.
• Test statistic. The test statistic is a t statistic (t) defined by the following equation.

  t = b1 / SE

  where b1 is the slope of the sample regression line, and SE is the standard error of the slope.
• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to thesignificance level, and rejecting the null hypothesis when the P-value is less than the significance level.

If we reject null hypothesis then we get B1 is not equal to zero