AP®StatisticsCollege BoardStudy GuidesUnits 6-9: InferenceInference for MeansConfidence Intervals for Differences in Population Means

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

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Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Two-sample t-interval for difference in population means

What is a confidence interval for the difference between two population means?

  • A confidence interval for the difference between two population means is

    • a symmetric range of values centered about the sample means difference

    • designed to capture the actual value of the difference between two population means

  • Different samples generate different confidence intervals

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

How do I calculate a confidence interval for the difference between two population means?

  • The confidence interval for the difference between two population means is given by

    • difference space between space sample space means space plus-or-minus open parentheses critical space value close parentheses open parentheses standard space error space of space sample space means space of space two space populations close parentheses

  • Where:

    • The difference between the two sample means is calculated from the samples 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 different the difference between two population means is likely to be from the difference between the two sample means, table row blank blank cell square root of s subscript A squared over n subscript A plus s subscript B squared over n subscript B end root end cell end table

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 difference between the sample means and the standard error of the difference of the sample means.

What are conditions for a confidence interval for a difference in population means?

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

    • the two samples are independent of each other

    • items in the samples (or experiments) 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 populations are approximately normally distributed

      • The distribution of both populations 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

    • 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 of space the space sample space means close parentheses

  • The confidence interval is

    • difference space in space sample space means 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 sizes increase!

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% confident that the interval from [lower limit] to [upper limit]...'

      • using the limits from the confidence interval

    • then end with it capturing the difference between the population means in context

      • e.g. 'captures the actual difference between the population means of the time taken by students from each school to run 100 m'

  • Confidence intervals for differences may have negative limits

    • This means that the difference, mu subscript 1 minus mu subscript 2, is negative

      • so mu subscript 1 less than mu subscript 2

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

  • If a population mean difference 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 difference is that value

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

  • Look out for confidence intervals for differences that contains zero

    • This means mu subscript 1 minus mu subscript 2 equals 0 so there is evidence to suggest mu subscript 1 equals mu subscript 2

Worked Example

Two independent farmers each claim to grow the longest eggplants. A random sample of 18 eggplants from the thousands grown at each farm are measured. The lengths of the eggplants on both farms is normally distributed. Those from farm A have a mean length of 8.1 inches with a standard deviation of 0.36 inches and the eggplants from farm B have a mean length of 7.9 inches with a standard deviation of 0.28 inches. Find the 95% confidence interval for mu subscript A minus mu subscript B.

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

The correct inference procedure is a two-sample t-interval with a 95% confidence level

  • The independence condition is satisfied, as

    • the samples are taken from different farms so they are independent

    • the samples of 18 eggplants were selected at random

    • the sample size, 18, is less than 10% of the population (thousands)

  • The distribution of lengths is normal

  • The sample size is small (n equals 18, which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Define the population parameters

Let mu subscript A be the mean length of the eggplants from farm A

Let mu subscript B be the mean length of the eggplants from farm B

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

table row cell n subscript A end cell equals 18 row cell x with bar on top subscript A end cell equals cell 8.1 end cell row cell s subscript A end cell equals cell 0.36 end cell end table table row cell n subscript B end cell equals 18 row cell x with bar on top subscript B end cell equals cell 7.9 end cell row cell s subscript B end cell equals cell 0.28 end cell end table

State the degrees of freedom (dof)

dof = 18 - 1 = 17

Using the t-table, find the t-value (critical value) for the sample mean, using dof = 17 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 = 2.110

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 open parentheses x with bar on top subscript A minus x with bar on top subscript B close parentheses plus-or-minus t asterisk times times square root of s subscript A squared over n subscript A plus s subscript B squared over n subscript B end root end cell row blank equals cell open parentheses 8.1 minus 7.9 close parentheses plus-or-minus 2.110 times square root of fraction numerator 0.36 squared over denominator 18 end fraction plus fraction numerator 0.28 squared over denominator 18 end fraction end root end cell end table

State the confidence interval

open parentheses negative 0.027 comma space 0.427 close parentheses

Explain the confidence interval in the context of the question

We can be 95% confident that the interval from -0.027 inches to 0.427 inches captures the actual value of the difference between the population means mu subscript A minus mu subscript B , of the lengths of eggplants from both farms

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    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.

    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.