Hypothesis Tests for Slopes of Regression Lines (College Board AP® Statistics)

t-test for slope of regression line

What is a t-test for the slope of a regression line?

• A t-test for the slope of a regression line is used to test whether the population slope, , of the population least-squares regression line, has changed

  • A random sample of observations from the population, with a sample least-squares regression line of is used to try to prove the case

What are the hypotheses for a t-test for a slope?

• The null hypothesis, , is the assumption that the population slope has not changed

  • e.g. The population slope has the fixed value

    • It is assumed to be correct, unless evidence proves otherwise

• The alternative hypothesis, , is how you think the population slope has changed

  • e.g. or or

What are the conditions for a t-test for a slope?

• When performing a t-test for a slope, you must show that it meets the following conditions:

  • The relationship between and must be linear

    • Look for randomness in a population residual plot

  • The standard deviation of the -values (responses), , cannot vary with

    • Check on a population residual plot that the lengths (vertical heights) of residuals stay roughly the same as increases horizontally

  • The residuals are independent

    • by verifying that data is collected by random sampling

    • or random assignment (in an experiment)

    • and, if sampling without replacement, showing that the sample size is less than 10% of the population size

  • For a given value of , the responses (-values) follow an approximate normal distribution

    • A population residual plot should show residuals evenly spread either side of the horizontal zero line

    • There should be more points in the inner horizontal band (either side of the zero line) and fewer points in the outer horizontal bands

  • If the sample size is , the distribution of responses (-values) should have no strong skew and no outliers

    • Look at a population residual plot to see if there is a bias (skew) to one side of the horizontal zero line, and look for outliers

    • If the distribution does have a skew, then the sample size must be

How do I calculate the standardized test statistic (t-value)?

• You need a measure of how far the sample slope is from the population slope

  • This is the standardized test statistic (in this case, called the t-value)

• The t-value, for the slope is given by:

  • 

  • where is the sample slope, is the population slope under the null hypothesis , and is the standard error of the sample slope given by

    • where is the sample size

    • and where

    • and

• The t-value shows how many standard errors the sample slope is from the population slope

How do I read output information on regression from a computer?

• The formulas for the standardized test statistic are complicated to use in practice

  • Computers are often used to calculate these quantities instead

• You may be given computer output information from a sample, as follows:

• The above computer output says

  • the sample least-squares regression line is

    • This gives the sample slope, , needed for

  • the standard error of the sample slope, , is 0.3421

    • This gives the standard error, , needed for

    • do not confuse with , which is an estimate for the standard deviation of population residuals (the standard error of the residuals)

How do I calculate the p-value?

• Work out the t-value (standardized test statistic)

• Find the appropriate number of degrees of freedom ('dof')

  • For a t-test for a slope this is always

    • If is known in , then

• Using the t-distribution table given to you:

  • find the row that corresponds to the dof

  • identify the t-value in the row that is closest to the calculated value

  • write down the value in the corresponding column header

    • this is the p-value

• Note that the p-value from the t-table is for one tail

  • You need to double this value for a two-tail test

How do I conclude a hypothesis test?

• Conclusions to a hypothesis test need to show two things:

  • a decision about the null hypothesis

  • an interpretation of this decision in the context of the question

• To make the decision, compare the p-value to the significance level

  • If then the null hypothesis should be rejected

  • If then the null hypothesis should not be rejected

• In a two-tailed test, remember to double the p-value and compare this to

What are the steps for performing a t-test for a slope on a calculator?

• When using a calculator to conduct a t-test for a slope, you must still write down all steps of the hypothesis testing process:

  • State the null and alternative hypotheses and clearly define your parameter

  • Describe the test being used and show that the situation meets the conditions required

  • Calculate the t-value and the degrees of freedom

  • Calculate the p-value using your calculator

    • select a one-sample t-test and enter the relevant summary statistics or data to generate the p-value

  • Compare the p-value to the significance level

  • Write down the conclusion to the test and interpret it in the context of the problem