AP STATISTICS LESSON 12 - 1

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AP STATISTICSLESSON 12 - 1

• INFERENCE FOR A POPULATION PROPORTION

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ESSENTIAL QUESTION What are the procedures
for creating significance tests and confidence
intervals for population proportion problems?

• Objectives
• To create confidence intervals for population
  proportions.
• To find significance for proportion populations.

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Introduction

• We often want to answer questions about the
  proportion of some outcome in a population, or to
  compare proportions across several populations.

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Population Proportion ProblemsPage 685

• Example 12.1 Risky Behavior in the Age of AIDS
  (estimating a single population proportion)
• Example 12.2 Does Preschool Make a Difference?
  (comparing two population proporations)
• Example 12.3 Extracurriculars and Grades
  (comparing more than two population proportions)

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Inference for a Population Proportion

• We are interested in the unknown proportion p
  of a population that has some outcome.
• For convenience, call the outcome we are
  looking for a success.

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Sample Proportion

• ? count of successes in the sample
• count of observations in the sample
• Read the sample proportion ? as p-hat.

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Conditions for Inference

• As always, inference is based on the sampling
  distribution of a statistic.
• The mean is p. That is, the sample proportion p
  is an unbiased estimator of the population
  proportion p. The standard deviation of p is v
  p(1-p)/n, provided that the population is at
  least 10 times as large as the sample. If the
  sample size is large enough that both np and n(1
  p ) are at least 10, the distribution of p is
  approximately normal.

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z Statistic

• z (p p)/ vp(1 p )/n
• The statistic z has approximately the standard
  normal distribution N(0,1) if the sample is not
  too small and the sample is not a large part of
  the population.

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Working Without p

• To test the null hypothesis Ho p p0 that the
  unknown p has a specific value po, just replace p
  by po in the z statistic and in checking the
  values of np and n(1 p).
• In a confidence interval for p, we have no
  specific value to substitute. In large samples,
  p will be close to p. So we replace p by p in
  determining the values of np and n(1 p). We
  also replace the standard deviation by the
  standard error of p
• SE vp(1 p)/n to get a confidence
  interval estimate zSE

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Conditions for Inference About a Proportion

• The data are an SRS from the population of
  interest.
• The population is at least 10 times as large as
  the sample.
• For a test Ho p po , the sample size n is so
  large that both npo and
• n(1 po) are 10 or more. For a confidence
  interval, n is so large that both the count of
  successes np and the count of the failures n( 1
  p ) are 10 or more.

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Example 12.4 Page 688Are the Conditions Met?

• The sampling design was in fact a complex
  stratified sample, and the survey used inference
  procedures for that design. The overall effect
  is close to an SRS, however.
• The number of adult heterosexuals
• (the population) is much larger than 10 times
  the sample size, n 2673

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Inference for a Population Proportion

• Draw an SRS of size n from a large population
  with unknown proportion p of success. An
  approximate level C confidence interval for p is
• p zv p(1 p ) / n
• Where z is the upper (1-C)/2 standard normal
  critical value. To test the hypothesis Ho p
  po compute the z statistic z (p po )/vpo(1
  po)/n

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Inference for Population Proportion (continued)

• In terms of a variable Z having the standard
  normal distribution, the approximate P-value for
  a test Ho against
• Ha p gt po is P(Z z )
• Ha p lt po is P(Z z )
• Ha p ? po is 2P(Z lzl )

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Example 12.5 Page 690 Estimating Risky Behavior

• The National AIDS Behavioral Surveys found
  that 170 of a sample of 2673 adult heterosexuals
  had multiple partners. That is, p 0.0636.
• A 99 confidence interval for the proportion
  p of all adult heterosexuals with multiple
  partners uses the standard normal critical value
  z 2.576 (use the bottom row of Table C for
  standard normal critical values)
• We are 99 confident that the percent of
  adult heterosexuals who had more than one sexual
  partner in the past year lies between about 5.1
  and 7.6

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Example 12.6 Page 691Binge Drinking in
College

• Binge drinking for men 5 or more drinks
  (women 4 or more drinks) on at lease one
  occasion within two weeks.
• In a representative sample of 140 colleges
  and 17,592 students (SRS), 7741 students
  identified themselves as binge drinkers.
• Does this constitute strong evidence that
  more than 40 of all college students engage in
  binge drinking?
• Answer
• The P-value tells us that there is virtually
  no change of obtaining a sample proportion as far
  away from0.40 as p 0.44. We reject H0 and
  conclude that more than 40 of U.S. college
  students have engaged in binge drinking.

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Example 12.7 Page 692Is That Coin Fair?

• A coin that is balanced should come up heads half
  the time in the long run. The French naturalist
  Count Buffon tossed a coin 4040 times and got
  2048 heads (p 0.5069)
• Is this evidence that Buffons coin was not
  balanced? (hint use the p-value for the
  two-sided test)

Answer We failed to find good evidence against
H0 p 0.5. We cannot conclude that H0 is true,
that is, that the coin is perfectly balanced.
NOTE The test of significance only shows that
the results of Buffons 4040 tosses cant
distinguish this coin from one that is perfectly
balanced. To see what values of p are consistent
with sample results, use a confidence interval.
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Example 12.8 Page 693Confidence Interval For p

• We are 95 confident that the probailiby of a
  head is between 0.4915 and 0.52223.
• The confidence interval is more informative
  than the text in Example 12.7.

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Choosing the Sample Size

• In planning a study, we may want to choose a
  sample size that will allow us to estimate the
  parameter within a given margin of error.
• m z v p(1 p )/ n
• Here z is the standard normal critical value
  for the level of confidence we want.
• Because the margin of error involves the
  sample proportion of success p, we need to guess
  this value when choosing n.
• Call our guess p. Here are two ways to get
  p.

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Ways to Get p

• Use a guess or p based on a pilot study or on
  past experience with similar studies. You should
  do several calculations that cover the range of
  p-values you might get.
• Use p 0.5 as the guess. The margin of error m
  is larger when
• p 0.5, so this guess is conservative in
  the sense that if we get other p when we do our
  study, we will get a margin of error smaller than
  planned.

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Sample Size for Desired Margin of Error

• To determine the sample size n that will yield
  a level C confidence interval for a population
  proportion p with a specified margin of error m,
  set the following expression for the margin of
  error to be less than or equal to m, and solve
  for n
• z vp(1 p) / n m
• Where p is a guessed value for the sample
  proportion. The margin of error will be less
  than or equal to m if you take the guess p to be
  0.5

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Choosing p

• The method for finding the guess p does not
  matter that much in most cases. The n you get
  doesnt change much when you change p as long as
  p is not too far from .5. So use the
  conservative guess p 0.5 if you expect the
  true p to be roughly between 0.3 and 0.7. If the
  true p is close to 0 or 1, using p as your guess
  will give a sample much larger than you need. So
  try to use a better guess from a pilot study when
  you suspect that p will be less than 0.3 or
  greater than 0.7.

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Example 12.9 Page 696Determining Sample Size
for Election Polling

• Find sample size for 2.5 margin of error (sample
  size n 1537),
• and
• for 2 margin of error (n 2041).