AP®StatisticsCollege BoardRevision NotesUnits 6-9: InferenceInference for MeansConfidence Intervals for Differences in Matched Pairs

Confidence Intervals for Differences in Matched Pairs (College Board AP® Statistics)

Revision Note

Naomi C

Author

Naomi C

Expertise

Maths

Paired t-interval

What is a confidence interval for differences in matched pairs?

  • A confidence interval for differences in matched pairs is

    • a symmetric range of values centered about the sample mean of the differences of the matched pairs

    • designed to capture the actual value of the population mean of the differences of the matched pairs

  • Different samples generate different confidence intervals

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

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

How do I calculate a confidence interval for differences in matched pairs?

  • The confidence interval for the population mean of the differences of the matched pairs is given by

    • Error converting from MathML to accessible text.

      • Where x with bar on top subscript d space is the mean of the differences between the matched pairs in the sample

      • Error converting from MathML to accessible text. is the standard deviation of the differences of the matched pairs in the sample

      • and Error converting from MathML to accessible text. is the standard error of the differences between the matched pairs in the sample

  • Where:

    • The mean of the differences in the sample 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 how big the difference is likely to be between

      • the mean of the differences between the matched pairs in the population

      • and the mean of the differences between the matched pairs in the sample

Exam Tip

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 mean of the differences of the matched pairs in the sample and the standard error of these differences.

What are conditions for a confidence interval for differences in matched pairs?

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

    • The two measures come from the same items within the population

    • 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 should be approximately normally distributed

      • The distribution needs to be approximately symmetric

      • There should be no outliers

    • If the population is very skewed, you can only do a t-test if n greater or equal than 30

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 the space difference space between space the space matched space pairs space in space the space sample close parentheses

  • The confidence interval is

    • mean space of space the space differences space between space the space matched space pairs space in space the space sample 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

Exam Tip

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 differences in matched pairs?

  • 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 difference in population means of the matched pairs in context

      • e.g. 'captures the actual population difference between the blood pressures before and after the medication was taken'

How do I use confidence intervals to justify a claim about differences in matched pairs?

  • If the mean of the differences between the matched pairs in the population 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 mean of the differences between the matched pairs in the population is that value

    • If it does not, the sample data does not provide sufficient evidence that the mean of the differences between the matched pairs in the population is that value

Worked Example

A random sample of 29 adults from a large town were asked to complete a puzzle, and the time it took each of them to complete it was recorded. A week later, the same group of adults was asked to complete the same puzzle. Their times were again recorded and the difference between their first and second attempts was calculated, with the time of the second attempt being subtracted from the first attempt, x subscript 1 minus x subscript 2. The mean of the differences between these attempts is 0.8 minutes and the standard deviation of the differences is 0.15 minutes. The differences between their times is assumed to be normally distributed.

The researchers use the sample to calculate a confidence interval for the actual difference in the population for the time taken between the first and second attempts. Given that the interval they calculate is (0.731 , 0.869), determine the level of confidence used.

State the type of test being used and verify the conditions for the test

The correct inference procedure is a paired t-interval

  • The data is paired

  • The independence condition is satisfied, as

    • the sample of 29 was selected at random

    • and the sample size, 29, is less than 10% of the population, implied by a 'large town'

  • The distribution of weights is approximately normally distributed

    • the sample size is small (n equals 29, which is less than 30), so the t-distribution can be used

Define the population parameter, mu subscript d

Let mu subscript d be the mean difference between the time taken to complete the puzzle on the first attempt and the time taken to complete the puzzle on the second attempt

List the number of data items in the sample, n, the mean of the differences of the matched pairs in the sample, x with bar on top subscript d, and the standard deviation of the differences in the sample, s subscript d

table row n equals 29 row cell x with bar on top subscript d end cell equals cell 0.8 end cell row cell s subscript d end cell equals cell 0.15 end cell end table

State the degrees of freedom (dof)

dof = 29 - 1 = 28

Using the given confidence interval (0.731, 0.869) and 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 critical value

You can calculate the critical value using either bound of the confidence interval

table row CI equals cell x with bar on top subscript d plus-or-minus t asterisk times times fraction numerator s subscript d over denominator square root of n end fraction end cell row cell 0.731 end cell equals cell 0.8 minus t asterisk times times fraction numerator 0.15 over denominator square root of 29 end fraction end cell row cell t asterisk times end cell equals cell open parentheses 0.8 minus 0.731 close parentheses times fraction numerator square root of 29 over denominator 0.15 end fraction end cell row cell t asterisk times end cell equals cell 2.477... end cell end table

Using the t-table, use the row for dof = 28 and find the t-value (critical value) closest to the value just calculated

closest t-value = 2.462

The associated column will indicate the confidence level

confidence level = 98%

Explain the confidence level in the context of the question

The researchers can be 98% confident that the interval from 0.731 minutes to 0.869 minutes captures the actual difference in the population for the time taken between the first and second attempts at the puzzle

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    Naomi C

    Author: Naomi C

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.