AP STATISTICS LESSON 10

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AP STATISTICSLESSON 10 2DAY 1

• TEST OF SIGNIFICANCE

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ESSENTIAL QUESTION What is a test of
significance and how are they used?

• Objectives
• To create tests of significance that will assess
  the evidence provided by data about some claim
  concerning a population.
• To draw conclusions from the tests of
  significance.

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Tests of Significance

• The second common type of inference, called
  test of significance, has a different goal
• To assess the evidence provided by data about
  some claim concerning a population.

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Example 10.8 Page 559 Im a Great Free-Throw
Shooter

• I claim to be an 80 free throw shooter. I
  make 8 of 20 free throws. Is this possible if my
  claim is true?
• Significance tests use an elaborate
  vocabulary, but the basic idea is simple
• An outcome that would rarely happen if a claim
  were true is good evidence that the claim is not
  true.

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Example 10.9 Page 560Sweetening Colas

• Significance test A test of significance
  asks does the sample results x 1.02 reflect a
  real loss of sweetness or could we easily get the
  outcome just by chance?
• Null hypothesis state the null hypothesis.
  The null hypothesis says that there is no effect
  or no change in the population. If the null
  hypothesis is true, the sample result is just
  chance at work.
• Ho µ 0
• We write Ho, read H-nought, to indicate the
  null hypothesis

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Example 10.9 (continued)

• The effect suspect is true, the alternative
  to no effect or no change, is described by
  the alternative hypothesis. We suspect that the
  cola does lose sweetness. In terms of the mean
  sweetness loss µ, the alternative hypothesis is
• Ha µ gt 0
• Suppose for the sake of argument that the
  null hypothesis is true, and that on the average
  there is no loss of sweetness.
• Is the sample outcome x 1.02 surprisingly
  large under that supposition? If it is, thats
  evidence against Ho and in favor of Ha?.

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How Does a Significance Test Work?

• A significance test works by asking how unlikely
  the observed outcome would be if the null
  hypothesis were really true?
• We measure the strength of the evidence against
  Ho by the probability under the normal curve in
  figure 10.10 to the right of the observed x.
  This probability is called the P-value.
• It is the probability of a result at least as
  far out as the result we actually got?
• The lower this probability, the more
  surprising our result, and the stronger the
  evidence against the null hypothesis.

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P-values

• Small P-values are evidence against Ho because
  they say that the observed result is unlikely to
  occur just by chance. Large P-values fail to
  give evidence against Ho.
• How small must a P-value be in order to
  persuade us? Theres no fixed rule. But the
  level 0.05 (a result that would occur no more
  than once in 20 tries just by chance) is a common
  rule of thumb.
• A result with a small P-value, say less than
  0.05 is called statistically significant.

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Outline of a Test

• Here is the reasoning of a significance test in
  outline form
• Describe the effect you are searching for in
  terms of a population parameter like the mean µ.
  (Never state a hypothesis of a sample statistic
  like x.)
• From the data, calculate a statistic like x that
  estimates the parameter. Is the value of this
  statistic far from the parameter value stated by
  the null hypothesis? If so, the data give
  evidence that the null hypothesis is false and
  that the effect you are looking for is really
  there.
• The P-value says how unlikely a result at least
  as extreme as the one we observed would be if the
  null hypothesis were true. Results with small
  P-value would rarely occur if the null hypothesis
  were true. We call such results statistically
  significant.