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Significance Test


Significance Test A claim is made. Is the claim true? Is the claim false? A test of signicance assesses the evidence found in the data against a null hypothesis in ... – PowerPoint PPT presentation

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Title: Significance Test

Significance Test
  • A claim is made.
  • Is the claim true?
  • Is the claim false?

  • A test of signicance assesses the evidence found
    in the data against a null hypothesis
  • in favor of an alternative hypothesis .

Null Hypothesis
  • The null hypothesis H0 is always of the form
  • H0
  • The null hypothesis says that "the claim is true
    and that what we observe in the
  • data happened by chance."

Alternative Hypothesis
  • The alternative hypothesis Ha always takes one of
    three forms
  • Two-sided Ha
  • One-sided on the high side Ha
  • One-sided on the low side Ha
  • Note 1 The form of the alternative hypothesis
    is problem dependent.
  • Note 2 The alternative hypothesis says "the
    claim is false and what we observe happened
    because the claim is false."

z Test for a Population Mean
  • Step 1 Write down the Null and Alternative
  • Null Hypothesis
  • Alternative Hypothesis

Step 2
Compute the test statistic (z-score)
Step 3
  • Compute the p- value
  • run a normalcdf
  • For a 2 sided test you will need to double this
  • The P-value is defined this way
  • the probability, computed assuming the null is
    true, that the test statistic would take a value
    as extreme or more extreme than that actually
    observed is called the P-value of the test.
  • The smaller the P-value, the stronger the
    evidence against the null hypothesis

Step 4
  • Compare p-value against the alpha level
  • -alpha level is a value that is too extreme to
    assume that the sample happened just by chance
  • -if p-value falls below the alpha level than the
    null is rejected
  • -if p-value is above the alpha level than we fail
    to reject the null
  • If the P-value is as small or smaller than alpha
    , we say that the data are statistically
    significant at that level.

  • Radon is a colorless, odorless gas that is
    naturally released by rocks and soils and may
    concentrate in tightly closed houses. Because
    Radon is slightly radioactive, there is some
    concern that it may be a health hazard. Radon
    detectors are sold to home owners worried about
    this risk. The detectors may be inaccurate. You
    placed 12 detectors in a chamber where they were
    exposed to 105 picocuries per liter (pCi/l) of
    Radon over three days. Here are the readings
    given by the detectors
  • 91.9 97.8 111.4 122.3 105.4 95.0
  • 103.8 99.6 119.3 104.8 101.7 96.6
  • Assume (unrealistically) that you know the
    standard deviation of readings for all detectors
    of this type is 9 and that the population
    of readings is approximately normal.

  • (i) Give a 95 confidence interval for the mean
    reading of this type of detector.

(ii) Is there significant evidence at the 5
level that the mean reading differs from the true
value of 105. State the null and alternative
hypotheses and conduct the significance test.
Assume the 12 detectors are an SRS. Already
stated that normally distributed. Ok to use
normal calculations.
  • With a P-value as high as 0.74 are sample has
    very good chance of occurring if the true mean is
    105. Therefore, we do not have enough evidence
    to reject the null and assume the mean reading
    could be around 105.
  • In other words we have no reason to think that
    the average value of the readings of all such
    detectors is not 105 and that what we observed,
    namely that x 104.13, did not happen by chance

Suppose that in a particular geographic region,
the mean and standard deviation of scores on a
reading test are 100 points, and 12 points,
respectively. Our interest is in the scores of 55
students in a particular school who received a
mean score of 96. We can ask whether this mean
score is significantly lower than the regional
mean that is, are the students in this school
comparable to a simple random sample of 55
students from the region as a whole, or are their
scores surprisingly low?
  • 10.38-40