Confidence Intervals for Population Means (College Board AP® Statistics)

One-sample t-interval for a mean

What is a confidence interval for a population mean?

• A confidence interval for a population mean is

  • a symmetric range of values centered about the sample mean

  • designed to capture the actual value of the population mean

• 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)

How do I calculate a confidence interval for a population mean?

• The confidence interval for a population mean is given by

  • 

• Where:

  • The sample mean is calculated from the sample or is given to you

  • The critical value is the relevant t-value

    • The critical value depends on the confidence level C%

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

What are conditions for a confidence interval for a population mean?

• When calculating a confidence interval, you must show that:

  • items in the sample (or experiment) must satisfy the independence condition

    • 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

  • the population is approximately normally distributed

    • 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

  • 

• 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 mean?

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

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

    • using the limits from the confidence interval

  • then end with it capturing the population mean in context

    • e.g. 'captures the actual population mean of the time taken by students in the school to run 100 m'

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

• If a population mean 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 mean is that value

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