Introduction to Confidence Intervals (College Board AP® Statistics)

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Introduction to confidence intervals

What is a confidence interval?

  • Recall that when a population parameter is unknown, a sample from the population is taken

    • An estimate of the population parameter can be made from the sample

      • e.g. the mean of a sample is an 'estimate' for the mean of the population

  • A confidence interval is

    • a symmetric range of values centered about an estimate from a sample

    • designed to capture the actual value of the population parameter

  • Different samples generate different confidence intervals

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

  • The population parameter is 'captured' if its actual value falls within a given confidence interval

What is the confidence level of a confidence interval?

  • Because we do not know the true value of the population parameter, we can only be, say, 95% sure that a confidence interval captures it

    • This percentage is called the confidence level

      • It represents the probability that a confidence interval captures the population parameter

  • Because different samples generate different confidence intervals

    • the confidence level can also be thought of as

      • the percentage of all possible confidence intervals (from all possible random samples of the same size, taken from the same population) that capture the population parameter

    • so if, say, 100 confidence intervals were generated from random samples of size n taken from the same population

      • you would expect, on average, 95 of them to capture the actual population parameter

  • Remember that the population parameter is a fixed (constant) value

    • It is the confidence intervals that change location, depending on the sample

      • A confidence interval either captures the population parameter or does not

Examiner Tips and Tricks

Be very careful with the wording in the exam! The confidence level is not the probability that the population parameter lies in an interval (as that sounds like the population parameter can move around), it is the probability than a confidence interval captures the population parameter.

How do I calculate a confidence interval?

  • A confidence interval has the general formula

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

  • The statistic is the estimate from the sample

  • The critical value depends on the confidence level, C%

    • e.g. the positive z-score that marks the middle C% of a standard normal distribution

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

    • This depends on the statistic

Examiner Tips and Tricks

The formula for confidence intervals (including a table of standard errors) is given in the exam.

What are conditions for a confidence interval?

  • Each type of confidence interval has its own set of conditions that must be met

  • A common condition is the independence condition which states that items in the sample (or experiment) must be independent

    • This is checked in two stages, first:

      • verify that data is collected by random sampling

      • or random assignment (in an experiment)

      • (This also allows the interval to be generalized to the population)

    • Second, if sampling without replacement:

      • verify that the sample size is less than or equal to 10% of the population size

      • Sometimes written n less or equal than 0.1 N

  • Another common condition is normality of the population or sampling distribution

    • The distribution needs to be approximately symmetric

    • There should be no outliers

Examiner Tips and Tricks

You must learn the conditions relevant to each type of confidence interval.

What is the margin of error?

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

    • From the formula, this means

      • margin space of space error space equals open parentheses critical space value close parentheses cross times open parentheses standard space error space of space statistic close parentheses

    • The confidence interval is

      • statistic 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

What affects the width of a confidence interval?

  • The confidence level affects the width

    • Increasing the confidence level will increase the width

    • Decreasing the confidence level will decrease the width

  • The size of the sample, n, affects the width

    • Increasing the sample size will decrease the width

    • Decreasing the sample size will increase the width

How do I interpret a confidence interval?

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

    • Start by saying 'we can be C% confident that...'

    • then say 'the interval from [lower limit] to [upper limit]'

      • using the limits from the confidence interval

    • then end with it capturing the population parameter in context

      • e.g. 'captures the actual value of the proportion of left-handed students in the school'

How do I use confidence intervals to justify a claim?

  • If a population parameter 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 parameter has that value

    • If it does not, the sample data does not provide sufficient evidence that the population parameter has that value

Examiner Tips and Tricks

In the exam, always word these conclusions in context (e.g. replace the word 'population parameter' with what it is).

How do I lay out a confidence interval question in the exam?

  • STEP 1
    Identify the appropriate inference procedure:

    • This is the type of confidence interval you want to create

      • e.g. 'The appropriate inference procedure is a one-sample z-interval for the proportion of left-handed students in the school'

      • Copy out the population parameter in context

  • STEP 2
    Verify any conditions relevant for that confidence interval

  • STEP 3
    Find the confidence interval

    • Write out statistic space plus-or-minus open parentheses critical space value close parentheses cross times open parentheses standard space error space of space statistic close parentheses

      • substituting in the correct expressions

    • Substitute numbers into the expressions above

      • e.g. 0.5 plus-or-minus 1.96 square root of fraction numerator 0.2 cross times 0.8 over denominator 200 end fraction end root

    • State the final interval

      • e.g. (0.445, 0.555)

  • STEP 4
    Conclusion:

    • 'We can be C% confident that the interval from [lower limit] to [upper limit] captures the actual value of the [population parameter in context]'

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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.