Tests About a Population Proportion

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SECTION 12.2

• Tests About a Population Proportion

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NOW WHAT?

• In this section we are interested in the
  unknown proportion, p of a population as opposed
  to the unknown mean of a population.
• Keep in mind, p will have an approximately normal
  distribution, so it is
• BACK TO THE WORLD OF z.

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Our z statistic

• We dont really know p for our standard
  deviation.
• So, when we do a test, replace p by p0.
• NOTE When we did confidence intervals, we used
  in place of p instead of p0

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Assumptions

• Data are an SRS from the population
• The population is at least ten times as large as
  the sample (which insures independence)
• A. For a significance test
• B. For a confidence interval

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The Steps for a One Proportion z-test

• State the hypothesis and name test
• H0 p p0
• Ha p , , or ? p0
• State and verify your assumptions
• Calculate the P-value and other important values
• Done in calculator or
• Using the formulas and tables
• State Conclusions (Both statistically and
  contextually)
• - The smaller the P-value, the greater the
  evidence is to reject H0

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Example

• A coin is tossed 4040 times. There were 2048
  heads. The sample proportion of heads is
• 2048/4040 0.5069
• Thats a bit more than one-half. Is this evidence
  that the coin was not balanced?

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Step 1Parameter

• The population for coin tossing contains the
  results of tossing a coin forever. The parameter
  p is the proportion of all tosses that lands
  heads up. The null hypothesis says that the coin
  is balanced. The alternative hypothesis is
  two-sided, because we did not suspect before
  seeing the data that the coin favored either
  heads or tails.

H0 p 0.5 Ha p ? 0.5
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Step 2Conditions

• SRSThe tosses we make can be considered an SRS
  from the population of all tosses.
• NormalitySince np04040(.5)2020
• and n(1-p0)4040(.5)2020 are both at least 10,
  we are safe using Normal calculations
• IndependenceSince we are sampling without
  replacement (?) we must have at least 40400
  tosses in our population. That isnt an issue.

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Step 3Calculations

• P-value 0.3783

Dont forget to draw your curve. Remember, use
p0 for your standard error calculations. Use
this standard error when drawing the curve.
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Step 4Interpretation

• A proportion of heads as far from one-half (.5)
  as this one would happen about 38 of the time by
  chance alone, if the coin is balanced.
• For this reason, we would fail to reject the null
  hypothesis.
• There is virtually no evidence that the coin is
  unbalanced.
• As a reminder, this is not evidence that the null
  hypothesis is true. It is still possible the
  coin is unbalanced, we just dont have strong
  enough evidence to convince anyone that it is
  unbalanced.

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Using a Confidence Interval

• For the example of the coin, it is possible that
  the confidence interval would be more meaningful
  than the significance test. A 95 confidence
  interval is
• (0.49152, 0.52235)
• We can see that 0.5 is plausible, but so are many
  higher proportions, including the proportion that
  we saw in our sample of 4040 tosses.

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Another Example

• Publishing scientific papers online is fast, and
  the papers can be long. Publishing in a paper
  journal means that the paper will live forever in
  libraries. The British Medical Journal combines
  the two it prints short and readable versions,
  with longer versions available online. It this
  OK with authors? The journal asked a random
  sample of 104 of its recent authors several
  questions. One question in the survey asked
  whether authors would accept a stronger move
  toward online publishing As an author, how
  acceptable would it be for us to publish only the
  abstract of papers in the paper journal and
  continue to put the full long version on our
  website? Of the 104 authors in the sample, 65
  said Not at all acceptable.
• Do the data provide good evidence that more than
  half of all authors feel that abstract-only
  publishing is not acceptable?

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Step 1Parameter

• The population of interest is all of the authors
  for this particular journal.
• The parameter is the proportion of these authors
  that disagree with the abstract-only printing of
  their articles
• The null hypothesis is that there will be an
  event split between those that oppose the
  abstract-only printing and those in favor.
• The alternative hypothesis is that more authors
  will be against the abstract-only printing.

H0 p 0.5 Ha p gt 0.5
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Step 2Conditions

• SRSThe chosen authors were a random sample but
  not necessarily an SRS of all authors from this
  journal.
• NormalitySince np0n(1-p0)52 are both at least
  10, we are safe using Normal calculations
• IndependenceSince we are sampling without
  replacement we must have at least 1040 authors
  for this magazine in the population.

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Step 3Calculations

• P-value 0.0054

Dont forget to draw your curve. Remember, use
p0 for your standard error calculations. Use
this standard error when drawing the curve.
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Step 4Interpretation

• Because of the small P-value, there is sufficient
  evidence to reject the null hypothesis.
• We can conclude that more than half of all
  authors from the British Medical Journal would be
  opposed to printing their articles in the
  abstract-only format.