Testing Hypotheses About Proportions

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Chapter 20

• Testing Hypotheses About Proportions

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Confidence Interval

• Confidence Interval for p

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Confidence Interval

• Plausible values for the unknown population
  proportion, p.
• We have confidence in the process that produced
  this interval.

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Inference

• Propose a value for the population proportion, p.
• Does the sample data support this value?

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Comparing Confidence Intervals with Hypothesis
Tests

• Confidence Interval A level of confidence is
  chosen. We determine a range of possible values
  for the parameter that are consistent with the
  data (at the chosen confidence level).

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Comparing Confidence Intervals with Hypothesis
Tests

• Hypothesis Test Only one possible value for the
  parameter, called the hypothesized value, is
  tested. We determine the strength of the
  evidence provided by the data against the
  proposition that the hypothesized value is the
  true value.

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Example - Hypothesis Test

• A law firm will represent people in a class
  action lawsuit against a car manufacturer only if
  it is sure that more than 10 of the cars have a
  particular defect.
• Population Cars of a particular make and model.
• Parameter Proportion of this make and model of
  car that have a particular defect.

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Example - Hypothesis Test

• Test a claim about the population proportion, p
• Start by formulating a hypotheses.
• Null Hypothesis
• H0 p 0.10
• Alternative Hypothesis
• HA p gt 0.10

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Parts to a Hypothesis Test

• Null Hypothesis (H0)
• What the model is believed to be
• H0 p p0
• Ex Fair coin H0 p 0.5

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Parts to a Hypothesis Test

• Alternative Hypothesis (Ha)
• Claim you would like to prove
• Ha p lt p0
• Ha p gt p0
• Ha p ? p0
• Ex Fair Coin? Ha p ? 0.5

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Writing a Hypotheses

• In the 1950s only about 40 of high school
  graduates went on to college. Has the percentage
  changed?
• H0 p .4 vs. Ha p ? .4
• Is a coin fair?
• H0 p .5 vs. Ha p ? .5
• Only about 20 of people who try to quit smoking
  succeed. Sellers of a motivational tape claim
  that listening to the recorded messages can help
  people quit.
• H0 p .2 vs. Ha p gt .2

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Writing a Hypotheses

• A governor is concerned about his negatives
  (the percentage of state residents who express
  disapproval of his job performance.) His
  political committee pays for a series of TV ads,
  hoping that they can keep the negatives below
  30. They will use follow-up polling to assess
  the ads effectiveness
• H0 p .3 vs. Ha p lt .3
• Coke will only market their new zero calorie soft
  drink only if they are sure that 60 percent of
  the people like the flavor
• H0 p .6 vs. Ha p gt .6

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Relationship Between H0 and Ha

• Law Order
• We assume people accused of a crime are innocent
  until proven guilty.
• H0 person is innocent
• You, as the prosecutor, must gather enough
  evidence to prove that the person accused is
  guilty beyond a shadow of a doubt.
• Ha person is not innocent.

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Example - Hypothesis Test

• Next, take a random sample and calculate
• The law firm contacts 100 car owners at random
  and finds out that 12 of them have cars that have
  the defect. Thus .12.
• Is this sufficient evidence for the law firm to
  proceed with the class action law suit?
• Ask How likely is it that our came from a
  population with a mean po ?
• If it is likely, then I have no reason to
  question the value for po.
• If it is unlikely, then we do have reason to
  question the value of po.

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Example - Hypothesis Test

• Check your assumptions!
• npo and nqo are greater than or equal to 10
• n is less than 10 of the population
• random sample
• independent values

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Example - Hypothesis Test

• Sampling Distribution of if Null Hypothesis
  is True

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Example - Hypothesis Test

• Sampling distribution of
• Shape approximately normal because 10 condition
  and success/failure condition satisfied.
• Mean p 0.10 (because we assume H0 is true)
• Standard Deviation

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Example - Hypothesis Test

• Calculate a Test Statistic

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Example - Hypothesis Test
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Example - Hypothesis Test
Use Z-Table
z 0.05 0.06 0.07 0.05 0.06
0.7486 0.07
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Example - Hypothesis Test
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Example - Hypothesis Test
Interpretation

• Getting a sample proportion of 0.12 or more will
  happen about 25 (P-value 0.25) of the time
  when taking a random sample of 100 from a
  population whose population proportion is p
  0.10.

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Example - Hypothesis Test
Interpretation

• Getting a value of the sample proportion of 0.12
  is consistent with random sampling from a
  population with proportion p 0.10. This sample
  result does not contradict the null hypothesis.
  The P-value is not small, therefore fail to
  reject H0.

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Parts to a Hypothesis Test

• P-value The probability of getting the observed
  statistic (i.e. ) or one that is more
  extreme given that the null hypothesis is true.
• Ha p lt po (one-sided test)
• p-value P(Z lt z)
• Ha p gt po (one-sided test)
• p-value P(Z gt z)
• Ha p ? po (two-sided test)
• p-value 2P(Z gt z)

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Ha p lt po p-value P(Z lt z)
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Ha p gt po p-value P(Z gt z) P(Z lt -z)
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Ha p ? po p-value 2P(Z gt z) 2P(Z lt -z)
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Parts to a Hypothesis Test

• Decision
• Small p-values mean there is evidence that null
  hypothesis is incorrect.
• Large p-values mean there is no evidence that
  null hypothesis is incorrect.
• What values are considered small or large?
• alpha level (significance level) a
• Typical values (0.01, 0.05, 0.10)

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Parts to a Hypothesis Test

• Decision (in terms of H0)
• Reject H0
• When p-value is smaller than a
• Enough evidence exists to say that H0 is most
  likely incorrect.
• Do not reject H0
• When p-value is larger than a
• Not enough evidence exists to say that H0 is
  incorrect.

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Parts to a Hypothesis Test

• Conclusion (in terms of Ha)
• If we reject H0, the conclusion would be
• There is evidence in favor of Ha
• If we fail to reject H0, the conclusion would be
• There is not enough evidence in favor of Ha

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Parts to a Hypothesis Test

• Decision
• Just remember one phrase If the p-value lt ?,
  reject H0
• Conclusion
• What have you decided about p?
• Stated in terms of problem

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Parts to a Hypothesis Test

• Step 1 Hypotheses
• Specify ?
• Step 2 Test Statistic
• Step 3 P-value
• Step 4 Decision Conclusion

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Example 1

• Many people have trouble programming their VCRs,
  so a company has developed what it hopes will be
  easier instructions. The goal is to have at
  least 90 of all customers succeed. The company
  tests the new system on 200 randomly selected
  people, and 188 of them were successful. Do you
  think the new system meets the companys goal?

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Example 1

• Step 1
• Population parameter of concern
• p proportion of people who successfully program
  their VCRs.
• Hypotheses
• H0 p 0.9
• Ha p gt 0.9
• We want to test this hypothesis at a ?.05 level

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Example 1

• Step 2
• Assumptions
• Random sample
• Independence
• npo 200(0.9) 180 gt 10
• nqo 200(0.1) 20 gt 10
• n 200 is less than 10 of the population size

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Example 1

• Step 2
• Model
• Test Statistic

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Example 1

• Step 3
• P-value
• P(Z gt z) P(Z gt 1.90) 0.0287
• If p-value lt ?, reject H0.
• Is there strong enough evidence to reject H0?
• If we want strong evidence (beyond a shadow of a
  doubt), ? should be small.

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Example 1

• Step 4
• Decision
• 0.0287 p-value lt ? 0.05, so reject H0.
• Conclusion
• There is evidence that the new system works in
  helping customers succeed in programming their
  VCRs.

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Example 1

• What is the interpretation of the p-value in the
  context of the problem?
• If the true proportion of people that can
  successfully program their VCR with the new
  instructions is 90, the probability of getting a
  sample proportion of 94 or one higher (more
  extreme) is about 2.9 (i.e. not very likely).

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Example 2

• In 1991, the state of New Mexico became concerned
  that their DWI rate was considerably above the
  national average. The national average that year,
  was .00809. Suppose they set up road blocks to
  allow them to randomly select drivers and record
  (and arrest) the number who were above the legal
  blood alcohol level. Out of a random sample of
  100,000 drivers, 2213 were above the limit (and
  subsequently arrested). Was there strong evidence
  that the DWI rate in New Mexico was higher than
  the national average?

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Example 2

• Step 1
• Parameter of interest
• p proportion of New Mexicans that have blood
  alcohol level above the limit.
• Hypotheses
• H0 p 0.00809
• Ha p gt 0.00809

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Example 2

• Step 2
• Check the necessary assumptions
• npo 100,000(0.00809) 809
• nqo 100,000(0.99191) 99191
• The population of New Mexico in 1991 was
  1,547,115. Our sample size of 100,000 is less
  than 10 of the population.
• Random sample
• independence

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Example 2

• Step 2
• Model
• Test Statistic

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Example 2

• Step 3
• P-value
• P(Z gt 49.61) P(Z lt -49.61) 0 (or lt 0.0001)
• Step 4
• Decision
• Use ? 0.05
• P-value lt ?, reject H0.

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Example 2

• Step 4
• Conclusion
• There is enough evidence to say that New Mexicos
  DWI is probably higher than the national average.
• Does driving in New Mexico cause you to be drunk?
• No, we are providing statistical inference based
  on data (evidence) gathered.

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Example 2

• What does the p-value mean in the context of this
  problem?
• If the true proportion of New Mexicans that have
  a blood alcohol level above the legal limit is
  0.809, the probability of getting a sample
  proportion of 2.2 or higher (more extreme)
  almost 0 (i.e. very unlikely).

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General Notes

• Always list both the null and alternative
  hypotheses for each problem.
• Remember that the null states a value for the
  population parameter p.
• The null arises from the context of the problem,
  not from the sample.
• We start by assuming that the null is true.
• If we find evidence, we can reject the null, but
  we never accept the null. We can fail to reject
  the null.
• The alternative states what your alternate
  assumptions is if you reject the null.

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General Notes

• Know how to determine whether you should use a
  one-sided or two-sided model.
• It depends upon how the question is worded.
• The z-statistic will be the same for each model,
  but the final p-value will change.
• Know the way to determine the p-value in each
  case.
• Always remember to interpret your conclusion in
  terms of the problem.
• State what the outcome is and what the likelihood
  of it occurring is.

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Example 3

• A large company hopes to improve satisfaction,
  setting as a goal that no more than 5 negative
  comments. A random survey of 350 customers found
  only 13 with complaints. Is the company meeting
  its goal?

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Example 3

• Step 1
• Population parameter of concern
• p proportion of dissatisfied customers
• H0 p 0.05
• Ha p lt 0.05

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Example 3

• Step 2
• Assumptions
• 350(0.05) 17.5
• 350(0.95) 332.5
• The company is large, so 350 is probably less
  than 10 of all of their customers
• Sample was random
• Customers are independent

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Example 3

• Step 2
• Model
• Test Statistic

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Example 3

• Step 3
• P-value P(Zlt-1.08) 0.1401
• Step 4
• Decision
• Since p-value 0.1401 gt a 0.05, fail to reject
  Ho
• Conclusion
• There is no evidence that the company is meeting
  its goal of receiving less than 5 negative
  comments.

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Example 3

• What is the interpretation of the p-value in the
  context of this problem?
• If the true proportion of customers that are
  dissatisfied is 5, the probability of getting a
  sample proportion of 3.7 or less (more extreme)
  is about 14.

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Example 4

• An airlines public relations department says
  that the airline rarely loses passengers
  luggage. It futher claims that on those
  occasions when luggage is lost, 90 is recovered
  and delivered to it owner within 24 hours. A
  consumer group who surveyed a large number of air
  travelers found that only 103 out of 122 people
  who lost luggage on that airline were reunited
  with the missing items by the next day. What do
  you think about the airlines claim? Use a 0.05

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Example 4

• Step 1
• Population parameter of concern
• p proportion of people who lost their luggage
  and had it returned within 24 hours
• HO p 0.9
• HA p lt 0.9

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Example 4

• Step 2
• Assumptions
• 122(0.9) 109.8
• 122(0.1) 12.2
• 122 is less than 10 of all people who have ever
  lost luggage on this airline.
• Random sample
• Independent values

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Example 4

• Step 2
• Model
• Test Statistic

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Example 4

• Step 3
• P-value

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Example 4

• Step 4
• Decision
• p-value 0.0192 lt a 0.05, reject HO
• Conclusion The population proportion of people
  who lost their luggage that have it returned
  within 24 hours on this airline is less than 90.
  The airlines claim is probably not true.

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Example 4

• What is the interpretation of the p-value in the
  context of this problem?
• If the true proportion of people who lost their
  luggage and had it returned within 24 hours is
  90, then the probability of getting a sample
  proportion of 84 or less (more extreme) is about
  1.9 (pretty unlikely).