Confidence Intervals for Slopes of Regression Lines (College Board AP® Statistics)

t-interval for slope of regression line

What is a confidence interval for the population slope?

• A confidence interval for the population slope is

  • a symmetric range of values centered about a sample slope

  • designed to capture the actual value of the population slope

• Different samples generate different confidence intervals

  • e.g. a sample slope of 5 may have a confidence interval of (4.5, 5.5)

  • while another sample slope of 5 may have a confidence interval of (4.1, 5.1)

How do I calculate a confidence interval for the population slope?

• The confidence interval for the population slope is given by

  • 

• Where:

  • The sample slope, , is calculated from the sample or is given to you

  • The critical value is the relevant t-value at degrees of freedom

    • The critical value depends on the confidence level C%

    • If is known in the population least-squares regression line, , then degrees of freedom are used

  • The standard error of the sample slope is an estimate of the population standard deviation from the data

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

• The formulas for regression with slopes 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 confidence interval

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

    • This gives the standard error, , needed for the confidence interval

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

What are the conditions for a confidence interval for the population slope?

• When calculating a t-interval for a population 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

What is the margin of error?

• The margin of error is the half-width of the confidence interval

  • 

• The confidence interval is

  • 

• The total width of a confidence interval is

• You may be given an interval and asked to calculate its margin of error

  • or another value, such as

    • This involves forming and solving an equation

How do I interpret a confidence interval for a population slope?

• You must conclude calculations of a confidence interval by referring to the context

  • Start by saying 'we can be C% confident that the interval from [lower limit] to [upper limit]...'

    • using the limits from the confidence interval

  • then end with it capturing the population slope in context

    • e.g. 'captures the actual population slope from the population least-squares regression line relating the time taken by students in the school to run 100 m to the score on their reaction test'

How do I use confidence intervals to justify a claim about a population slope?

• If a population slope is claimed to be a specific value

  • check if that value lies in your confidence interval

• If it does, the sample data provides sufficient evidence that the population slope is that value

  • If it does not, the sample data does not provide sufficient evidence that the population slope is that value

• Look out for confidence intervals that include zero, 0

  • A slope of zero suggests that does not depend in any way on