Estimating a population proportion

1
Estimating a population proportion

• ASW, 6.3, 7.6, 8.4

Economics 224 notes for October 20, 2008
2
Normal approximation to binomial (ASW, 6.3)

• If a probability experiment has n independent
  trials with p as the probability of success and
  1-p as the probability of failure, the
  probabilities of the number of successes, x, have
  a binomial probability distribution.
• The probabilities for x, where x 0, 1, 2, 3,
  ... , n are given by the expression
• For small n, it is not too difficult to obtain
  the values of f(x) with a calculator or from
  binomial tables.
• For large n, the calculation is more difficult if
  a computer program is not available.
• Fortunately, when n is large, the normal
  probability distribution can be used to
  approximate the binomial probabilities.

3
Which normal distribution?

• For the binomial probability distribution, the
  mean and standard deviation, respectively, are
• If np 5 and n(1-p) 5, the normal distribution
  with the above mean and standard deviation
  provides a reasonable approximation to the
  binomial probabilities (ASW, 243).
• When calculating these, there is a continuity
  correction factor (ASW, 243) that must be used.
  For example, the probability of obtaining exactly
  4 successes would be the area under the normal
  curve between 3.5 and 4.5.
• The larger the value of n, the more closely the
  normal distribution approximates the binomial
  probabilities.

4
Population proportion p

• When conducting research about a population,
  researchers are often more interested in the
  proportion of a population with a particular
  characteristic, rather than the number of
  population elements with the characteristic.
• Proportion of population who support the
  Liberals.
• Proportion of manufactured objects that are
  defect free.
• Proportion of employees with extended health care
  plans.
• Percentage of the labour force that is
  unemployed.
• In each of these situations, the actual number of
  population elements with the characteristic will
  vary with the sample size. But the aim of
  obtaining samples is to estimate the proportion,
  or percentage, of the population with the
  characteristic.
• Let the proportion of a population with a
  particular characteristics be represented by p.

5
Terminology and notation for proportions

• p is the proportion of a population with a
  particular characteristic.
• Draw a random sample of size n elements from the
  population that contains N elements.
• Let x be the number of sample elements with the
  characteristic.
• Define the sample proportion as where
• That is, is the proportion of elements of
  the sample of size n that have the
  characteristic.

6
Sampling distribution of p

• If samples of size n are drawn from a population
  with proportion p having a particular
  characteristic, the sample proportion will
  differ from sample to sample. Some samples will
  have a larger proportion of sample elements with
  the characteristic and some will have a smaller
  proportion. The distribution of when there
  is repeated sampling is termed the sampling
  distribution of .
• If the sample size n is only a small proportion
  of the population size N, the sampling
  distribution of has a binomial distribution
  with a mean of p and a standard deviation of
• See ASW, 279-280 for these results.

7
Normal approximation for a proportion

• Recall that a binomial variable x has a mean of µ
  np with variance s2 np(1-p).
• For a binomial variable x/n, where x is
  divided by n, it should make sense that the mean
  and standard deviation of x divided by n produce
  a mean of µ p and a standard deviation
• for x/n.
• If np 5 and n(1-p) 5, the normal distribution
  provides a reasonable approximation to the
  binomial probabilities, so the distribution of
  the sample proportion is approximated by the
  normal distribution with the above mean and
  standard deviation (ASW, 280-281).
• From this, the probability of different levels of
  sampling error for the sample proportion can be
  calculated (ASW, 281-282).

8
Estimating a population proportion

• Let p be the proportion of a population with a
  particular characteristic. If a large random
  sample of n elements of the population is drawn
  from this population, the sample proportion
  is approximated by a normal distribution
  with mean and standard deviation, respectively,
  being
• Since the population proportion is unknown and is
  being estimated, the above standard deviation is
  also unknown. However, the sample proportion
  often is a reasonable estimate of p, so in
  practice the mean and standard deviation,
  respectively, of the distribution of the sample
  proportion are
• From the results on the previous slides, the
  margin of error

9
Margin of error for a proportion

• From the previous slides, it follows that (1
  a)100 of the random samples are associated with
  the following margin of error E when estimating a
  population proportion
• This result holds only if the sample size n is
  large, that is np 5 and n(1-p) 5, so the
  binomial probabilities are approximated by areas
  under the normal distribution.

10
Interval estimate for a population proportion p

• When n is large, the (1-a)100 confidence
  interval for estimating p, the proportion of a
  population with a particular characteristic, is
• where is the sample proportion and
  x is the number of sample elements with the
  characteristic.
• For this interval estimate, large n means
• For smaller n, the interval will be wider than
  given by this formula.

11
Example of opinion polling - I

• From the October 6, 2008 example of opinion polls
  prior to the November 2003 Saskatchewan
  provincial election, what is the margin of error
  for the Cutler poll?
• What is the interval estimate for the percentage
  of decided voters who say they will vote NDP?
• Use the 95 level of confidence in each case.

12
Percentage of respondents, votes, and number of
seats by party, November 5, 2003 Saskatchewan
provincial election
Sources CBC Poll results from Western Opinion
Research, Saskatchewan Election Survey for The
Canadian Broadcasting Corporation, October 27,
2003. Obtained from web site http//sask.cbc.ca/r
egional/servlet/View?filenamepoll_one031028,
November 7, 2003. Cutler poll results
provided by Fred Cutler and from the Leader-Post,
November 7, 2003, p. A5.
13
Example of opinion polling - II

• For the Cutler poll, n 773 and the conditions
  for a large sample size appear to hold. Using
  even the smallest value for the sample proportion
  reported (other at 2 or 0.02),
• Given this large n, the sample proportion is
  approximated by a normal distribution. At 95
  confidence level, the Z value is 1.96 and the
  margin of error is
• In this case, a value of 0.5 is used for the
  estimate of the sample proportion, since this
  produces the widest possible margin of error.

14
Example of opinion polling - III

• For the Cutler poll, the margin of error is plus
  or minus 0.035 or 3.5 per cent, with 95
  confidence. This means that with a sample of
  size n 773, the estimate of the proportion of
  the population who support any political party
  may be incorrect by as much as 3.5 percentage
  points in 95 out of 100 samples.
• Each public opinion poll should provide an
  estimate of the margin of error when reporting
  poll results. The margin of error is the amount
  E by which the sample proportion differs from the
  population proportion, plus a confidence level.
• For purposes of generating this margin of error
  that applies to any characteristic, use
  and this will provide an upper bound for the
  estimated margin of error.

15
Example of opinion polling - IV

• For the 95 confidence interval for the estimate
  of the proportion who support a party, note that
  the sample of decided voters is only 84 of the
  773 (16 were undecided) so that the actual
  sample size was n 0.84 x 773 649.
• For the NDP, the sample proportion is 0.47 and
  the conditions for large sample size are met, so
  the normal distribution can be used. At 95
  confidence, Z 1.96 and the interval is
• and the 95 interval estimate for the proportion
  who support the NDP is from 0.432 to 0.508.
  Note that this interval includes the actual
  proportion p 0.445 who supported the NDP in the
  election.

16
Sample size for a proportion

• For confidence level (1-a)100 and margin of
  error E, the required sample size is determined
  by solving the following expression for n.
• This gives the formula for sample size

17
Estimating sample size

• In the formula for sample size required for
  estimating a proportion, the value of the sample
  proportion is unknown. ASW (315) revise the
  formula to use a planning value p giving the
  formula
• When using the formula, if you let p 0.5, this
  produces the maximum possible value for n for any
  given E and a.
• If you consider it possible that the population
  proportion differs considerably from p 0.5, say
  p ? 0.2 or p 0.8, then use one of the
  guidelines in ASW (315).

18
Example of sample size for a proportion

• What sample size would be required to obtain an
  estimate of the proportion of University of
  Regina students who use Regina Transit to travel
  to the University, accurate to within 5
  percentage points, with 90 confidence?
• For this question, neither the sample nor
  population proportion are known so use a planning
  proportion of p 0.5. E 0.05 and Z 1.645.
  The required sample size is
• A random sample of n 271 UR students will give
  at least the precision necessary, and perhaps
  even greater precision.
• Assume that sampling method produces a random
  sample. If N 12,000, the sample is 2.3 of N,
  so the sample size is a small proportion of the
  population size.

19
Notes about sample size for estimating a
population proportion

• Random sample of a population.
• If the sample size is a small proportion of the
  population size (less than 5-10 of population),
  then it does not matter how large the population
  is, the required n is independent of population
  size.
• This formula is especially useful, since it does
  not require knowledge of the population
  variability. If p 0.5 is used in the above
  formula, the sample size will be more than
  sufficient to achieve the required margin of
  error with the specified level of confidence.
• Not too many nonsampling errors such as poorly
  constructed questions, nonresponse, refusals,
  etc.
• For more complex sampling procedures, consult a
  text on sampling procedures.

20

• Monday, Oct. 20 we will discuss the above
  slides and then have some time for review.
• Tuesday, Oct. 21, 330 430 p.m. Optional
  review period with your two instructors. CL232.
• Wednesday, Oct. 22, 230 345 is the midterm.
  You are permitted to bring a text, photocopies of
  the tables (normal, t, binomial), and one extra
  sheet. Make sure you bring a calculator. No
  communication with other individuals inside or
  outside of the classroom using electronic
  devices.
• The midterm covers the topics discussed in class
  to October 20, that is, the assigned sections of
  chapters 1-8 of the text and any additional
  materials discussed in class.
• We are hoping to have Assignment 3 graded and
  available to pick up at the Tuesday review
  session. Answers will be posted on UR Courses
  some time on Tuesday.