AP®StatisticsCollege BoardRevision NotesUnits 6-9: InferenceInference for ProportionsConfidence Intervals for Population Proportions

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

Revision Note

Mark Curtis

Expertise

Maths

One-sample z-interval for population proportion

What is a confidence interval for the population proportion?

  • A confidence interval for the population proportion is

    • a symmetric range of values centered about a sample proportion

    • designed to capture the actual value of the population proportion

  • Different samples generate different confidence intervals

    • e.g. a sample proportion of 0.3 may have a confidence interval of (0.25, 0.35)

    • while another sample proportion of 0.3 may have a confidence interval of (0.29, 0.31)

How do I calculate a confidence interval for the population proportion?

  • The confidence interval for the population proportion is given by

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

  • Where:

    • The sample proportion, p with hat on top, is calculated from the sample

    • The critical value is the relevant z-value

      • The critical value depends on the confidence level C%

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

      • given by square root of fraction numerator p with hat on top open parentheses 1 minus p with hat on top close parentheses over denominator n end fraction end root

      • where n is the size of 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 sample proportion and the standard error of the sample proportion.

What are the conditions for a confidence interval for the population proportion?

  • When calculating a z-interval for a population proportion, you must show that it meets the following conditions:

    • 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 sample size is large enough such that the sampling distribution of p with hat on top is approximately a normal distribution

      • by verifying that the expected number of successes, n p with hat on top, and expected number of failures, n open parentheses 1 minus p with hat on top close parentheses, are both at least 10:

      • n p with hat on top greater or equal than 10

      • n open parentheses 1 minus p with hat on top close parentheses greater or equal than 10

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 sample space proportion close parentheses

  • The confidence interval is

    • sample space proportion 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

      • If no value of p with hat on top is known, use p with hat on top equals 0.5 (as this gives the upper bound of n)

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 a population proportion?

  • 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 population proportion in context

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

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

  • If a population proportion 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 proportion is that value

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

Worked Example

In a college of 600 students, the principal takes a random sample of 50 students and finds out that 26 of these students walk to school. Calculate a 95% confidence interval for the proportion of students in the college who walk to school.

Answer:

Define the population parameter, p

Let p be the proportion of students who walk to school at the college

Calculate the proportion of students in the sample of 50 who walk to school

p with hat on top equals 26 over 50 equals 0.52

State the type of interval being used and verify that the conditions for the interval are met

The correct inference procedure is a one-sample z-interval for the population proportion at a 95% confidence level

  • The independence condition is satisfied, as

    • the sample of 50 students was selected randomly by the principal

    • and the sample size, 50, is less than 10% of the population of the school, 600 (50 < 60), which is required as sampling was conducted without replacement

  • The sample size is large enough for the sampling distribution of the sample proportion to be approximately a normal distribution, because

    • n p with hat on top equals 50 times 0.52 equals 26 greater or equal than 10

    • n open parentheses 1 minus p with hat on top close parentheses equals 50 times open parentheses 1 minus 0.52 close parentheses equals 24 greater or equal than 10

List the sample size, n, the sample proportion, p with hat on top, and the standard error of the sample proportion, square root of fraction numerator p with hat on top open parentheses 1 minus p with hat on top close parentheses over denominator n end fraction end root

table row n equals 50 row cell p with hat on top end cell equals cell 0.52 end cell row cell square root of fraction numerator p with hat on top open parentheses 1 minus p with hat on top close parentheses over denominator n end fraction end root end cell equals cell square root of fraction numerator 0.52 open parentheses 1 minus 0.52 close parentheses over denominator 50 end fraction end root equals 0.070654... end cell end table

Find the z-score (critical value) for a confidence level of 95%, e.g. from the tables

Remember that a confidence level of 95% is 5% in both tails combined, so use 2.5% for a single tail in the table
(Alternatively, the row for t subscript infinity in the t-tables are z-scores, together with the corresponding 'Confidence level C' shown below)

z-score = 1.960

Using the formula given to you in the exam, 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 p with hat on top plus-or-minus z times square root of fraction numerator p with hat on top open parentheses 1 minus p with hat on top close parentheses over denominator n end fraction end root end cell row blank equals cell 0.52 plus 1.960 times 0.070654... end cell end table

State the confidence interval

open parentheses 0.382 comma space 0.658 close parentheses

Explain the confidence interval in the context of the question

We can be 95% confident that the interval from 0.382 to 0.658 captures the actual value of the proportion of students in the college who walk to school

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    Mark Curtis

    Author: Mark Curtis

    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.