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

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

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)

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

  • The confidence interval for the population slope is given by

    • sample space slope space plus-or-minus open parentheses critical space value close parentheses open parentheses standard space error space of space sample space slope close parentheses

  • Where:

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

    • The critical value is the relevant t-value at n minus 2 degrees of freedom

      • The critical value depends on the confidence level C%

      • If alpha is known in the population least-squares regression line, y with hat on top equals alpha plus beta x, then n minus 1 degrees of freedom are used

    • The standard error of the sample slope, s subscript b, is an estimate of the population standard deviation from the data

Examiner Tips and Tricks

The general formula for confidence intervals (including a table of standard errors) is given in the exam: statistic space plus-or-minus open parentheses critical space value close parentheses open parentheses standard space error space of space statistic close parentheses.

You will need to apply it appropriately using the sample slope and the standard error of the sample slope.

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:

Predictor

Coef

SE Coef

T

P

Constant

33.243

4.436

7.49

0.000

[x-variable name]

1.1356

0.3421

3.32

0.004

S = 7.35829

R-Sq = 35.2 %

R-Sq (adj) = 32.1 %

  • The above computer output says

    • the sample least-squares regression line is y with hat on top equals 33.243 plus 1.1356 x

      • This gives the sample slope, b equals 1.1356, needed for the confidence interval

    • the standard error of the sample slope, s subscript b, is 0.3421

      • This gives the standard error, s subscript b equals 0.3421, needed for the confidence interval

      • do not confuse s subscript b with s equals 7.35829, which is an estimate for the standard deviation of population residuals (the standard error of the residuals)

Examiner Tips and Tricks

Make sure you know how to find the standard error of the sample slope from any computer output given in the exam (do not use the row for the 'constant' term!).

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 x and y must be linear

      • Look for randomness in a population residual plot

    • The standard deviation of the y-values (responses), sigma subscript y, cannot vary with x

      • Check on a population residual plot that the lengths (vertical heights) of residuals stay roughly the same as x 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 x, the responses (y-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 n less than 30, the distribution of responses (y-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

    • margin space of space error space equals open parentheses critical space value close parentheses open parentheses standard space error space of space sample space slope close parentheses

  • The confidence interval is

    • sample space slope plus-or-minus margin space of space error

  • The total width of a confidence interval is 2 cross times margin space of space error

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

    • or another value, such as n

      • This involves forming and solving an equation

Examiner Tips and Tricks

You need to know that the width of a confidence interval increases as the confidence level increases, whereas it decreases as the sample size increases!

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 y does not depend in any way on x

Worked Example

An ecologist takes a random sample of 26 ferns, measuring their distances from a volcanic crater, x kilometers, and their heights, y centimeters. The results from a computer analysis of the sample are shown below.

Predictor

Coef

SE Coef

T

P

Constant

33.243

4.436

7.49

0.000

Distance

1.1356

0.3421

3.32

0.004

S = 7.35829

R-Sq = 35.2 %

R-Sq (adj) = 32.1 %

Construct a 95% confidence interval for the slope of the population regression line relating distance from the volcanic crater to fern height. You may assume all conditions for inference are met.

Answer:

State the type of test being used and verify that the conditions for the test are met

The correct inference procedure is a t-interval for a slope with a 95% confidence level

It is assumed in the question that all conditions for inference are met

Define the population parameter, beta, for which you are constructing an interval

Let beta be the slope of the population regression line relating distance from the volcanic crater, x kilometers, to fern height, y centimeters

List the sample size, n, the sample slope, b (from the table), and the standard error of the sample slope, s subscript b (from the table)

table row n equals 26 row b equals cell 1.1356 end cell row cell s subscript b end cell equals cell 0.3421 end cell end table

State the degrees of freedom, using n minus 2

dof = 26 - 2 = 24

Using the t-table, find the t-value (critical value) for the sample slope using 24 degrees of freedom and a confidence level of 95%

Remember that a confidence level of 95% is 5% in both tails combined, so use 2.5% for a single tail in the table (the row 'Confidence level C' at the bottom of the t-tables helps)

t-value = 2.064

Using the formula from the formula sheet, Confidence space interval equals statistic plus-or-minus open parentheses critical space value close parentheses space open parentheses standard space error space of space statistic close parentheses, calculate the confidence interval

table row CI equals cell b plus-or-minus t asterisk times times s subscript b end cell row blank equals cell 1.1356 plus-or-minus 2.064 times 0.3421 end cell end table

State the confidence interval

open parentheses 0.430 comma space 1.842 close parentheses

Explain the confidence interval in the context of the question

We can be 95% confident that the interval from 0.430 to 1.842 captures the actual value of the slope of the population regression line relating distance from the volcanic crater, x kilometers, to fern height, y centimeters

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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.