Testing Hypotheses

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Unit 15

• Testing Hypotheses
• Proportions

2
We Want Answers!

• People want answer to questions like
• Has the presidents approval rating changed since
  last month?
• Has teenage smoking decreased in the past 5
  years?
• Is global temperature increasing?
• Did the Super Bowl ad we bought actually increase
  sales?

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Hypotheses and Legal System

• The process of testing a hypothesis in statistics
  is very similar to the jury system.
• To prove someone guilty, we start by assuming
  they are innocent.
• We retain that hypothesis until the facts make it
  impossible beyond a reasonable doubt.

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Statistical Hypotheses

• In statistics we also begin by assuming a
  hypothesis is true
• Next we consider the data. Does it support the
  hypothesis or is it surprising?
• Statisticians measure how surprised they are
  using a p-value.
• The p-value measures the likelihood the
  hypothesis is true.

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Making Decisions

• If the p-value is large we retain the hypothesis
  we started with
• If the p-value is small we reject the hypothesis.
• We never actually prove a hypothesis is true.
  high p-values just lend it support.
• We also never prove it is wrong .. small
  p-values just suggest it is very suspicious.

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How Unlikely is Unlikely?

• Some researchers advocate 1 time out of 20 (.05)
  and others 1 time out of 100 (.01)
• Ultimately you have to determine that for
  yourself based on the nature of your research.
• There is no question in the smoking study we
  would reject the null hypothesis since the
  p-value was considerably below .01.

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Smoking

• National data in the 1960s showed that about 44
  of the adult population never smoked cigarettes.
• In 1995 a national health survey interviewed 881
  adults and found that 52 had never smoked
• Does this provide evidence of a change in smoking
  among adults?

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How Do We Test This?

• What is the hypothesis we want to test?
• Remember, we must assume that since the smoking
  percentage was 44 in the 1960s that nothing has
  changed since then.
• Our starting hypothesis, called the null
  hypothesis, denoted Ho, says that the percent of
  adults smoking in 1995 is still 44.

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Null Hypothesis

• The general form of a null hypothesis is
• Ho parameter hypothesized value
• In the smoking study Ho p .44
• This says the smoking rate in 1995 is still 44
• What would convince you it has changed?
• How much different from 44 would a sample
  proportion have to be?

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Think z-scores!

• We know sample proportions are normally
  distributed and that a z-score measures how many
  standard deviations from the true proportion, p,
  a given sample proportion, is.
• So lets calculate
• and see what happens

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Could This Have Happened?

• The z-score would be
• How likely is it to observe a value at least 4.78
  standard deviations from the mean?
• Normalcdf(4.78,99) .00000088
• Very unlikely!

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What Happens When Ho is Rejected?

• We better have an alternative in mind!
• The alternative hypothesis, denoted HA, specifies
  an interval of values
• HA contains the values of the parameter we accept
  if we reject the null hypothesis.
• For the smoking study HA would most likely be
• HA p gt .44 since given the bad press about
  smoking, more people are likely to avoid smoking.

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Alternative Alternatives

• Is there a difference in the percent of people
  who prefer Coke and Pepsi
• Two Sided Alternative Ho p.5, HA
• Here we are interested in the possibility that
  the percent that prefer Coke could be either more
  or less than .5, so the p-value is split between
  the two tails.

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One-Sided Alternative

• In the smoking study, researchers were only
  interested in the possibility that the percent of
  non-smokers has increased. The alternative
  hypothesis would be
• HA p gt .44
• And the p-value would all be concentrated in the
  upper tail

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How Do We Know?

• The decision to use a one or two tailed
  alternative hypothesis is based on the reason for
  conducting the study.
• If we are interested in determining if a
  difference exists, then its two-tailed.
• If we are interested in determining if there is
  an increase or decrease, then its one-tailed
• Look for clues in how the problem is stated

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Three Steps

• Testing a hypothesis about a population
  proportion involves 4 steps.
• State both the null and alternative hypotheses
• Calculate
• Obtain the p-value for the test
• Clearly state your conclusion and if rejecting
  the null hypothesis quote the p-value

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Football

• During the 2000 season, the home team won 128 of
  the 240 regular season National Football League
  games. Is this strong evidence of a home field
  advantage?
• Step 1 State the null and alternative hypotheses
• HO p .5
• HA p gt .5

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Next

• Step 2 Calculate the z-score for the test
• Step 3 Evaluate the p-value
• Normalcdf(1.033,99) .15. This is not less than
  .05.
• Step 4 State a conclusion
• Fail to reject HO. There is no evidence of a home
  field advantage.

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Create a CI

• A 95 confidence interval for the proportion of
  games won at home is
• .47 lt p (games won at home) lt .60
• With 95 certainty we state that in the National
  Football League, the home team wins between 47
  and 60 of the time. Since 50 is included in
  this interval we have no evidence of a home field
  advantage.

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On TI 83/84

• Select 51-PropZTest from the STAT TESTS menu
• Specify the hypothesized proportion
• Enter the observed value of x
• Enter the sample size
• Specify the form of the alternative hypothesis
• Calculate

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Twins

• In 2001 a national vital statistics report
  indicated that about 3 of all births produced
  twins. Data from a large city hospital found only
  7 sets of twins born to 469 teenage girls. Does
  this suggest that mothers under 20 are less
  likely to have twins?

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Solution

• Ho p .03, HA p lt .03
• z -1.91
• p- value .0278 (less than .05)
• Reject Ho. Teenage mothers are less likely than
  the general population to have twins p lt .03.
• 95 CI .004 lt p (twins) lt .026
• 95 confident that between .4 and 2.6 of
  teenage mothers have twins

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Overweight

• In a 2002 Gallop poll, 40 of all Americans
  considered themselves very or somewhat
  overweight. In a recent sample of 800 Americans,
  352 considered themselves very or somewhat
  overweight. Is the percent of Americans who
  consider themselves overweight today different
  than it was in 2002?

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Solution

• Ho p .4 HA
• Z2.309 p-value .0209 (less than .05)
• Reject Ho. The proportion of Americans who
  consider themselves overweight is more than 40
  (p lt .021).
• .41 lt p (overweight today) lt .47 is a 95 CI. We
  are 95 confident that between 41 and 47 of
  Americans consider themselves overweight today.