Introduction to Confidence Intervals (College Board AP® Statistics)

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)

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

• 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 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

How do I calculate a confidence interval?

• A confidence interval has the general formula

  • 

• 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

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

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

  • The distribution needs to be approximately symmetric

  • There should be no outliers

What is the margin of error?

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

  • From the formula, this means

    • 

  • 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

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, , 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

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

    • substituting in the correct expressions

  • Substitute numbers into the expressions above

    • e.g.

  • 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]'