Hypothesis

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Chapter 8

• Hypothesis
• Testing

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An example

• Suppose that we want to compare the crime rate in
  Detroit with the crime rate in the rest of the
  country.
• Is there more or less crime in Detroit than the
  national average?

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Null Hypothesis

• The hypothesis that we put to the test is called
  the null hypothesis, symbolized H0.
• The null hypothesis usually states the situation
  in which there is no difference (the difference
  is null) between populations.

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Alternative Hypothesis

• The alternative hypothesis, symbolized Ha, is the
  opposite of the null hypothesis.
• The alternative hypothesis is also identified as
  the research hypothesis, or the hunch that the
  investigator wants to test.

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Example

• Suppose that a new pesticide has been developed.
  The manufacturer claims that it will kill more
  than 60 of the ants it comes in contact with.
  100 ants are obtained and each is put in a dish.
  The chemical is applied and after 1 hour, 62 of
  the ants have died.

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• Is the pesticide really more than 60 effective
  in killing ants? The data seems to suggest so
  since 62/100 62 is more than 60 but there is
  variability in the data and 62 may not be
  statistically greater than 60.

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• H0 and Ha are opposites of each other and they
  are statements about parameter(s).
• For the ants example, if we let ? be the
  proportion of ants killed, we can write
• H0 ? 0.6 and Ha ? gt 0.6

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Type of tests

• right-tailed test
• left-tailed test
• two-tailed test

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Other Examples

• H0 ? 0.2 and Ha ? lt 0.2
• H0 ? 0.7 and Ha ? ? 0.7
• H0 µ 3 and Ha µ lt 3
• H0 µ -1 and Ha µ ? -1
• H0 µ 0 and Ha µ gt 0
• These all involve parameters such as µ and ?.
• Also, H0 always gets some form of equality
• (, or ).

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What are H0 and Ha ?

• Claim The mean IQ score of statistics students
  is more than 110.
• Claim The percentage of people that watch 60
  minutes each night is at most 25.

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Formulate hypothesis

• A conjecture is made that the mean starting
  salary for computer science graduates is 30,000
  per year. The researcher believes that it is
  greater than 30,000. Formulate the null and
  alternative to evaluate the claim.

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Test statistic

• A test statistic, i.e. a statistic from the
  sample for testing the null hypothesis. The
  choice of statistic depends on the parameter
  being tested.

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• Determine the sampling distribution of the test
  statistic under the assumption that Ho is true.
  Then it is possible to determine what values of
  the test statistic seem reasonable for rejection
  of the null hypothesis.

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• The rejection region consists of those values of
  the test statistic that will lead to the
  rejection of the null hypothesis

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p-Value

• The p-Value is the probability (computed when Ho
  is assumed to be true) of observing a value of
  the statistic at least as extreme as that given
  by the actual observed data. The smaller the
  p-value, the stronger is the evidence against the
  null hypothesis.
• It depends on the right-tailed, left-tailed and
  two-tailed test.

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p-Value

• It depends on the right-tailed, left-tailed and
  two-tailed test.
• Left tailed test p-value P(ZltZobs)
• Right tailed test p-value P(ZgtZobs)
• Two tailed test p-value 2P(ZgtZobs)

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P-value for right tailed test
p-value P(ZgtZobs)
For both, the p-value is the area to the right of
the TS.
Zobs
Zobs
Here, the p-value is less than a.
Here, the p-value is greater than a.
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P-value for left tailed test
p-value P(ZltZobs)
Zobs
Zobs
For both, the p-value is the area to the left of
the TS.
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P-value for Two Sided Test
Ha p ? p0
p-value 2P(Z gt Zobs)
P-value/2
P-value/2
Zobs
Zobs
For both, half of the p-value is the area to the
right of the Zobs. Zobs gt0
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P-value for Two Sided Test
Ha p ? p0
p-value 2P(Z gt Zobs)
P-value/2
P-value/2
TS
TS
For both, half of the p-value is the area to the
left of the Zobs. Zobslt0
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Example (486)

• The researcher believes that the mean capita
  income of the country residents is greater than
  15,000. Suppose the researcher selects a random
  sample of 100 county residents and finds that
  their average per capita income is 16,200.
  Suppose we know ?4,000, compute the p-value. Zobs

H0 ?15000 H1 ?gt15000
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Level of Significance ?

• The size of the rejection region, called the
  level of significance (denoted by ?), determines
  how small the p-value should be before we reject
  the null hypothesis.
• If p-value of is less than ?, we have
  significant evidence to reject H0 in favor of Ha.
  If p-value of is larger than ?, we will say we
  havent significant evidence to reject H0, so we
  will accept H0.

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Procedure for Testing Hypothesis

• Formulate null hypothesis Ho and alternative Ha.
  
• Decide on an appropriate test statistic.
• Determine the sampling distribution of the test
  statistic under the assumption that the null
  hypothesis is true.
• Set ?. Calculate the p-value and determine
  whether it is sufficiently small to reject the
  null hypothesis.
• Interpret the results.

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Example 8.4 (Page 489)

• The scores on a college placement exam in
  mathematics are assumed to be normally
  distributed with a mean of 70 and a standard
  deviation of 18. The exam is given to a random
  sample of 50 high school seniors who have been
  admitted to college. Their average score on the
  exam was 67. Is the evidence sufficient to
  suggest that the population mean score is lower
  than or equal to 70?

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Our decision

• will be either
• Reject H0 meaning we feel that we have
  sufficient evidence for Ha.
• Fail to Reject H0 were not conceding to accept
  H0 just yet.

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Errors

• Its not guaranteed that well make the correct
  decision. The only way that could happen is if
  we knew what the true population parameter is.

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4 Possibilities
OUR DECISION
THE TRUTH
Fail to Reject H0 (H0 is TRUE)
The ones in green are correct decisions No
Error. The ones in red are errors.
H0 is TRUE
Reject H0 (H0 is FALSE)
Fail to Reject H0 (H0 is TRUE)
H0 is FALSE
Reject H0 (H0 is FALSE)
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Types of Errors

• Type I Error Rejecting H0 is false when in fact
  H0 is true. The probability of committing a Type
  I error is denoted as ?.
• Type II Error Failing to reject H0 when in fact
  is H0 false. The probability of committing a Type
  II error is denoted by ?.
• On the previous page, which ones do these
  correspond to?

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• In the court system, we test
• H0Guilty vs. Ha Innocent
• What would be a Type I and Type II error
• here?

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• a is used in making a decision.
• ß is used in comparing tests.
• We preset a before collecting the data.
• Common values are 0.01, 0.05, 0.1.

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One Sided Tests (Right and Left)
Ha p gt p0
Ha p lt p0
a
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Two Sided Test

• All tests use the same test statistic.
• For all tests, Reject H0 when the test statistic
  is in the rejection region.

Ha p ? p0
a/2
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8.2 Testing a Population Proportion (continued)

• We wish to test
• Ho ?.60 Ha ?gt.60
• Suppose that the drug company investigates 200
  cases and finds that the new drug is effective in
  134 cases. Is this enough to reject Ho and say
  that ?gt.60?

What is p-value?
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Large-Sample Test of a Population Proportion

• Assumption ngt30
• Testing statistic

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• If Ha ?lt.60 What is p-value?

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• If Ha ??.60 What is p-value?

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Equivalence of Confidence Intervals and
Two-tailed Tests

• The null hypothesis Ho ??0 versus alternative
  Ha ???0 is rejected at an ? level of
  significance if and only if the hypothesized
  value falls outside a
• (1-?)100 confidence interval for ?.

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8.3 Testing a Population Mean µ

• The same general principles apply as they
• did for tests about p.
• We need a test statistic.
• We need to know the distribution of the test
  statistic.
• Compute the p-value and compare it to a.

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• If s is knownor ngt30, using Z-test. The test
  statistic is
• Use a Z (standard normal)
• to obtain p-values and
• critical values.

• If s is unknown, using T-test. The test statistic
  is
• Use a t distribution with n-1
• Degrees of freedom obtain
• p-values and critical values.

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• Whether you have a test about p or µ,
• you always reject H0 if p-value lta.

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Example 8.8

• To justify raising its rates, an insurance
  company claims that the mean medical expense for
  all middle-class families is at least 700 per
  year. A survey of 100 randomly selected
  middle-class families found that the mean
  medical expense for the year was 670 and the
  standard deviation was 140. Assuming that the
  tails of the distribution of medical expenses are
  not usually long, is there any evidence that the
  insurance company is misinformed?

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Example 8.4

• UM president Mary Sue Coleman claims that the
  percentage of UM students who like math is larger
  than that of MSU which is given by 0.6. MSU
  president Lou Anna doesnt agree. So I take a
  random sample of 80 from UM, find that only 35
  students who like math. Test Marys claim?

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End
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Example 5.4

• UM president Mary Sue Coleman claims that the
  percentage of UM students who like math is larger
  than that of MSU students . MSU president Lou
  Anna doesnt agree. So I take a sample of 100
  from MSU 80 from UM, find that 70 and 49 students
  respectively who like math. D? (nMSU 50 nUM
  60)