## Hypothesis Test for Slopes

Video from https://www.youtube.com/watch?v=q_ma0FtFp04

**Overview:**

This test tells you the probability that the slope in your linear regression model will be zero. If the slope were zero, then there would be no relationship between the x and y values.

**Standard Error:**

There are only three aspects of a scatter plot that affect the standard error of the regression slope:

1) Spread around the line:

2) Spread of x-values:

3) Sample size:

*n*

You want se to be low, sx to be large, and a larger sample size.

Student's t-model equation is:

This equation uses n-2 degrees of freedom.

The null hypothesis is that the slope equals zero. The alternative hypothesis is that the slope does not equal zero.

A scatterplot of the data should be approximately linear; there should be no pattern in the residuals or the variability of the residuals; the residuals should be approximately normal.

**Hypotheses:**The null hypothesis is that the slope equals zero. The alternative hypothesis is that the slope does not equal zero.

**Conditions:**A scatterplot of the data should be approximately linear; there should be no pattern in the residuals or the variability of the residuals; the residuals should be approximately normal.

**Example 1:**Using the table above, is there a significant linear relationship between annual bill (constant) and home size? Use a 0.05 level of significance.

1. First, state the hypothesis

H0: The slope of the regression line is equal to zero.

Ha: The slope of the regression line is not equal to zero.

Remember: If the relationship between home size and electric bill is significant, the slope will not equal zero.

2. State your alpha level: 0.05

3. Calculate the p-value of the slope: p=0.01

4. Since 0.01 is less than 0.05, you can reject the null hypothesis

5. Restate your conclusion: With a p-value of 0.01, which is less than , there is sufficient evidence to conclude that there is a relationship between annual bill and home size.

The table below gives the average weekly study time and the average test score of 20 randomly selected students. Is there sufficient evidence to conclude that there is a relationship between study times and test scores?

1. First, state the hypothesis

H0: The slope of the regression line is equal to zero.

Ha: The slope of the regression line is not equal to zero.

Remember: If the relationship between home size and electric bill is significant, the slope will not equal zero.

2. State your alpha level: 0.05

3. Calculate the p-value of the slope: p=0.01

4. Since 0.01 is less than 0.05, you can reject the null hypothesis

5. Restate your conclusion: With a p-value of 0.01, which is less than , there is sufficient evidence to conclude that there is a relationship between annual bill and home size.

**Example 2**The table below gives the average weekly study time and the average test score of 20 randomly selected students. Is there sufficient evidence to conclude that there is a relationship between study times and test scores?