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AP®StatisticsCollege BoardStudy GuidesUnit 5: Sampling DistributionsSampling DistributionsSampling Distributions for Sample Proportions

Sampling Distributions for Sample Proportions (College Board AP® Statistics): Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Sampling distributions for sample proportions

What is the population proportion?

  • In a population of individuals (or items), there may be a certain percentage of them who have a particular quality

    • e.g. 15% of students at a school are left-handed

  • In statistics an individual that has the particular quality is referred to as a success

  • The proportion of a population is the percentage of successes, written as a decimal

    • e.g. the proportion of left-handed students in the school is 0.15

  • The symbol p is used for the population proportion

    • e.g. p equals 0.15

Examiner Tips and Tricks

You may have to work out the population proportion yourself.

For example if a school has 1000 students and 150 of them are left-handed, then the population proportion of left-handed students is 150 over 1000 equals 0.15.

What is a sample proportion?

  • If a random sample of n individuals is taken from a population with population proportion p

    • the random sample will have its own proportion of successes

      • which may not be exactly equal to p

  • The symbol p with hat on top is used for the sample proportion

    • e.g. in a sample of 10 students, 2 were left-handed

      • so p with hat on top equals 2 over 10 equals 0.2

What is the sampling distribution for sample proportions?

  • In a sample of size n taken from the population

    • let X count the number of successes in the sample

      • so X is the number of successes in n trials

      • and each trial is either a success or a failure

  • X follows a binomial distribution with probability of success p

    • where p is the population proportion

  • The sample proportion, p with hat on top, is given by

    • p with hat on top equals X over n

      • The number of successes in the sample divided by the total number of individuals in the sample

  • If the sample size is large enough such that the conditions

    • n p greater or equal than 10

    • and n open parentheses 1 minus p close parentheses greater or equal than 10 are satisfied

    • then p with hat on top will follow:

      • an approximate normal distribution

      • with mean p

      • and standard deviation square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root

    • This is the sampling distribution for the sample proportion

What else should I know about the sampling distribution for sample proportions?

  • You need to know that

    • The standard deviation square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root assumes sampling was done with replacement

      • If sampling without replacement, make sure the sample size is less than 10% of the population size to be able to use square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root

      • otherwise the standard deviation will be smaller

    • The standard deviation square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root depends on n

      • A larger sample size gives a smaller standard deviation

    • You can use the approximate normal distribution to calculate probabilities involving sample proportions, p with hat on top

      • Its standardized z-statistic is fraction numerator p with hat on top minus p over denominator square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root end fraction

      • p, the population proportion, will be given in the question

    • If the sample size is not large enough (i.e. the two conditions are not satisfied) then the sampling distribution is not approximately normal

      • but the mean and standard deviation formulas still hold

Examiner Tips and Tricks

The mean, p, and the standard deviation, square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root, are given in the exam under 'Sampling distributions for proportions', in the row called 'For one population'.

Worked Example

In a particular country, 40% of the population live by the coast. A sample of 50 people from the country were surveyed to see if they lived by the coast or not.

Find the probability that less than half of those surveyed live by the coast.

Answer:

Write down the population proportion, p

p equals 0.4

Check if n p greater or equal than 10 and n open parentheses 1 minus p close parentheses greater or equal than 10 are satisfied, where n is the sample size, 50

50 cross times 0.4 equals 20 greater or equal than 10 and 50 cross times open parentheses 1 minus 0.4 close parentheses equals 30 greater or equal than 10

These conditions mean that the sampling distribution of the sample proportion can be treated as approximately normal with mean p and standard deviation square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root

Substitute p equals 0.4 and n equals 50 into square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root to find the standard deviation

table row cell square root of fraction numerator p open parentheses 1 minus p close parentheses over denominator n end fraction end root end cell equals cell square root of fraction numerator 0.4 cross times open parentheses 1 minus 0.4 close parentheses over denominator 50 end fraction end root end cell row blank equals cell square root of fraction numerator 0.4 cross times 0.6 over denominator 50 end fraction end root end cell row blank equals cell 0.06928203... end cell end table

If less than half of those surveyed live by the coast, the sample proportion, p with hat on top, is less than 0.5

P open parentheses p with hat on top less than 0.5 close parentheses

Use a normal distribution with mean 0.4 and standard deviation 0.06928203... from above to work out this probability

First calculate the z-score of 0.5

fraction numerator 0.5 minus 0.4 over denominator 0.06928203... end fraction equals 1.443...

Then find P open parentheses Z less than 1.443... close parentheses e.g. using the normal tables

P open parentheses Z less than 1.443... close parentheses equals 0.9251

The probability that less than half of those surveyed live by the coast is 0.9251

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.