Hypothesis Tests

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Chapter 8

• Hypothesis Tests

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Hypothesis Testing

• We now begin the phase of this course that
  discusses the highest achievement of statistics.
• Statistics, as the analytic branch of science,
  has provided scientists with a tool that, since
  the early part of the last century, has made
  possible many of the huge achievements of that
  century.
• You are aware, from science classes, of terms
  like conjecture, hypothesis, theory, and law.
• It is at the stage of hypothesis that most
  scientific research is done.
• Until now, you may not have been aware of the
  important role of statistics in this process. A
  whole new world is about to open up to you.

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What is a Hypothesis?

• In scientific parlance, a hypothesis is an
  educated guess about something research may
  reveal, or a potential answer to a question that
  is being investigated.
• In science, this is often called the research
  hypothesis and is a statement of the anticipated
  conclusion of the experiment.
• In the statistical analysis of an experiment, we
  state two hypotheses The null hypothesis and
  the alternative hypothesis.

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The Alternative Hypothesis

• The alternative hypothesis usually corresponds to
  the scientists research hypothesis.
• It is often a statement of what one hopes to
  prove in the experiment.
• In statistics, the alternative hypothesis may be
  written symbolically. Its name will be Ha or H1
  (in case of multiple hypotheses, they can be
  numbered)

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The Null Hypothesis

• The null hypothesis is a statement that expresses
  the conclusion if the experiment doesnt prove
  anything.
• It is often the status quo, what is currently
  accepted, or a standard we hope to beat.
• The null hypothesis is given the symbolic name H0
  (h-naught).

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Clear Thinking

• There are strong philosophical reasons for
  stating hypotheses in this way.
• Science progresses by proposing new ideas, which
  are tested, and accepted only if there is
  sufficient evidence to support the new idea over
  the old.
• The effect is to prevent science from going off
  on wild tangents. The benefit of the doubt goes
  to currently accepted beliefs, which ensures some
  stability and enforces a standard of proof for
  new ideas.

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Courtroom Analogy

• An important analogy is found in the American
  system of justice Innocent until proven guilty.
• Here, the H0 is innocence. That is what will be
  accepted in the event that evidence is
  insufficient or inconclusive.
• Ha is guilt. If evidence is sufficient (beyond a
  reasonable doubt), the alternative will be
  accepted.
• Note that a conviction is a conclusion that Ha is
  true, and thus a rejection of the innocence
  hypothesis, but an acquittal is not a declaration
  of innocence, only a conclusion that there is
  insufficient evidence to convict.
• We avoid saying accept H0, since H0 was assumed
  to begin with. We prefer to say either that we
  reject H0 or do not reject H0. It is OK to
  say accept Ha.

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Example Testing Problems

• In the previous chapter, we discussed estimating
  parameters.
• For example, use a sample mean to estimate µ,
  giving both a point estimate and a CI.
• Now we take a different approach. Suppose we
  have an existing belief about the value of µ.
  This could come from previous research, or it
  could be a standard that needs to be met.
• Examples
• Previous corn hybrids have achieved 100 bu/acre.
  We want to show that our new hybrid does better.
• Advertising claims have been made that there are
  20 chips in every chocolate chip cookie. Support
  or refute this claim.

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Stating the Null Hypothesis

• We start with a null hypothesis.
• The null hypothesis is denoted by H0 µµ0 where
  µ0 corresponds to the current belief or status
  quo.
• Example
• In the corn problem, if our hybrid is not better,
  it doesnt beat the previous yield achievement of
  100 bu/acre. Then we have H0 µ100 or possibly
  H0 µ100.
• In the cookie problem, if the advertising claims
  are correct, we have H0 µ20 or possibly H0
  µ20.
• Notice the choice of null hypothesis is not based
  on what we hope to prove, but on what is
  currently accepted.

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Stating the Alternative

• The alternative hypothesis is the result that you
  will get if your research proves something is
  different from status quo or from what is
  expected.
• It is denoted by Ha µ?µ0. Sometimes there is
  more than one alternative, so we can write H1
  µ?µ0, H2 µgtµ0, and H3 µltµ0.
• In the corn problem, if our yield is more than
  100 we have proved that our hybrid is better, so
  the alternative Ha µgt100 is appropriate.

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Stating the Alternative

• For the cookie example, if there are less than 20
  chips per cookie, the advertisers are wrong and
  possibly guilty of false advertising, so we want
  to prove Ha µlt20.
• A jar of peanut butter is supposed to have 16 oz
  in it. If there is too much, the cost goes up,
  while if it is too little, consumers will
  complain. Therefore we have H0 µ16 and Ha
  µ?16.
• From these examples, we can see that some tests
  focus on one direction and some do not.

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Comparison with Confidence Intervals

• In a confidence interval, our focus is to provide
  an estimate of a parameter.
• A hypothesis test makes use of an estimate, such
  as the sample mean, but is not directly concerned
  with estimation.
• The point is to determine if a proposed value of
  the parameter is likely to be untrue.

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Test of the Mean, s Known

• The null hypothesis is initially assumed true.
• It states that the mean has a particular value,
  µ0.
• Therefore, it follows that the distribution of
  x-bar has the same mean, µ0.
• The logic goes something like this. If we take a
  sample, we get a particular sample mean. If the
  null hypothesis is true, that mean is not likely
  to be far away from the hypothesized mean. It
  could happen, but its not likely. Therefore, if
  the sample mean is too far away, we will
  suspect something is wrong, and reject the null
  hypothesis.
• The next slide shows this graphically.

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Comments on the Graph

• What we see in the previous graph is the idea
  that lots of sample means will fall close to the
  true mean. About 68 fall within one standard
  deviation. There is still a 32 chance of
  getting a sample mean farther away than that.
  So, if a mean occurs more than one standard
  deviation away, we may still consider it quite
  possible that this is a random fluctuation,
  rather than a sign that something is wrong with
  the null hypothesis.

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More Comments

• If we go to two standard deviations, about 95 of
  observed means would be included. There is only a
  5 chance of getting a sample mean farther away
  than that. So, if a far-away mean occurs (more
  than two standard deviations out), we think it is
  more likely that it comes from a different
  distribution, rather than the one specified in
  the null hypothesis.

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Choosing a Significance Level

• The next graph shows what it means to choose a 5
  significance level.
• If the null hypothesis is true, there is only a
  5 chance that the standardized sample mean will
  be above 1.96 or below -1.96.
• These values will serve as a cutoff for the test.
• We are dealing only with cases where the sample
  mean can be assumed normal.

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Decision Time

• We have already shown that we can use a
  standardized value instead of to decide when
  to reject. We will call this value Z, the
  standard normal test statistic.
• The criterion by which we decide when to reject
  the null hypothesis is called a decision rule.
• We establish a cutoff value, beyond which is the
  rejection region. If Z falls into that region,
  we will reject Ho.
• The next slide shows this for a.05.

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One-tailed Tests

• Our graphs so far have shown tests with two
  tails.
• We have also seen that the alternative hypothesis
  could be of the form H2 µgtµ0, or H3 µltµ0.
• These are one-tailed tests. The rejection region
  only goes to one side, and all of a goes into one
  tail (it doesnt split).

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Making Mistakes

• Hypothesis testing is a statistical process,
  involving random events. As a result, we could
  make the wrong decision.
• A Type I Error occurs if we reject H0 when it is
  true. The probability of this is known as a, the
  level of significance.
• A Type II Error occurs when we fail to reject a
  false null hypothesis. The probability of this is
  known as ß.
• The Power of a test is 1-ß. This is the
  probability of rejecting the null hypothesis when
  it is false.

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Classification of Errors
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Two important numbers

• The significance level of a test is a, the
  probability of rejecting Ho if it is true.
• The power of a test is 1-ß, the probability of
  rejecting Ho if it is false.
• There is a kind of trade-off between significance
  and power. We want significance small and power
  large, but they tend to increase or decrease
  together.

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Steps in Hypothesis Testing

• State the null and alternative hypotheses
• Determine the appropriate type of test (check
  assumptions)
• Define the rejection region
• Calculate the test statistic
• State the conclusion in terms of the original
  problem

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p-Value Testing

• Say you are reporting some research in biology
  and in your paper you state that you have
  rejected the null hypothesis at the .10 level.
  Someone reviewing the paper may say, What if you
  used a .05 level? Would you still have
  rejected?
• To avoid this kind of question, researchers began
  reporting the p-value, which is actually the
  smallest a that would result in a rejection.
• Its kind of like coming at the problem from
  behind. Instead looking at a to determine a
  critical region, we let the estimate show us the
  critical region that would work.

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How p-Values Work

• To simplify the explanation, lets look at a
  right-tailed means test. We assume a
  distribution with mean µ0 and we calculate a
  sample mean.
• What if our sample mean fell right on the
  boundary of the critical region?
• This is just at the point where we would reject
  H0.
• So if we calculate the probability of a value
  greater than , this corresponds to the
  smallest a that results in a rejection.
• If the test is two tailed, we have to double the
  probability, because marks one part of the
  rejection region, but its negative marks the
  other part, on the other side (other tail).

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Using a p-Value

• Using a p-Value couldnt be easier. If plta, we
  reject H0. Thats it.
• p-Values tell us something about the strength
  of a rejection. If p is really small, we can be
  very confident in the decision.
• In real world problems, many p-Values turn out to
  be like .001 or even less. We can feel very good
  about a rejection in this case. However, if p is
  around .05 or .1, we might be a little nervous.
• When Fischer originally proposed these ideas
  early in the last century, he suggested three
  categories of decision
• p lt .05 ? Reject H0
• .05 p .20 ? more research needed
• p gt .20 ? Accept H0