Chapter 12 Tests of Hypotheses Means

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Chapter 12 Tests of Hypotheses Means

• 12.1 Tests of Hypotheses
• 12.2 Significance of Tests
• 12.3 Tests concerning Means
• 12.4 Tests concerning Means(unknown variance)
• 12.5 Differences between Means
• 12.6 Differences between Means(unknown variances)
  
• 12.7 Paired Data

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12.1 Tests of Hypotheses

• In a law case, there are 2 possibilities for the
  truthinnocent or guilty
• Evidence is gathered to decide whether to convict
  the defendant. The defendant is considered
  innocent unless proven to be guilty beyond a
  reasonable doubt. Just because a defendant is
  not found to be guilty doesnt prove the
  defendant is innocent. If there is not much
  evidence one way or the other the defendant is
  not found to be guilty.

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For Statistical Hypothesis Testing

• H0Null hypothesis (innocent)
• Held on to unless there is sufficient evidence
  to the contrary
• HAAlternative hypothesis (guilty)
• We reject H0 in favor of HA if there is enough
  evidence favoring HA

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12.1 Tests of Hypotheses

• Distribution(s) or population(s)
• Parameter(s) such as mean and variance
• Assertion or conjecture about the population(s)
  statistical hypotheses
• 1. About parameter(s) means or variances
• 2. About the type of populations normal ,
• binomial, or

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Example 12.1

• Is a coin balanced?
• This is the same as to ask if p0.5
• Is the average lifetime of a light bulb equal to
  1000 hours?
• The assertion is µ1000

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Null Hypotheses and Alternatives

• We call the above two assertions
• Null Hypotheses
• Notation H0 p0.5 and H0µ1000
• If we reject the above null hypotheses,
• the appropriate conclusions we arrive are
• called alternative hypotheses
• HA p?0.5 HA µ?1000

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Null Hypothesis vs Alternative

• H0 p0.5 vs HA p?0.5
• H0µ1000 vs HA µ?1000
• It is possible for you to specify other
  alternatives
• HA pgt0.5 or HA plt0.5
• HA µgt1000 or HA µlt1000

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12.2 Significance of Tests

• A company claims its light bulbs last on average
  1000 hours. We are going to test that claim.
• We might take the null and alternative hypotheses
  to be
• H0µ1000 vs HA µ?1000
• or may be
• H0µ1000 vs HA µlt1000

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Mistakes or errors

• Law caseconvict an innocent defendant or fail
  to convict a guilty defendant.
• The law system is set up so that the chance of
  convicting an innocent person is small. Innocent
  until proven guilty beyond a reasonable doubt.

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Two Types of Errors in statistical testing

• Type I error -- reject H0 when it is true
  (convict innocent person)
• Type II error -- accept H0 when it is not true
  (find guilty person innocent)

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Statistical hypotheses are set up to

• Control type I error
• ?P(type I error)
• P(reject H0 when H0 true)
• (a small number)
• Minimize type II error
• ?P(type II error)
• P(accept H0 when H0 false)

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Control Types of Errors

• In practice, ? is set at some small values,
  usually 0.05
• If you want to control ? at some small values,
  you need to figure out how large a sample size
  (n) is required to have a small ? also.
• 1- ? is called the power of the test
• 1- ?PowerP(reject H0 when H0 false)

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Example 12.2

• Xbreaking strength of a fish line, normal
  distributed withs0.10.
• Claim mean is ?10
• H0 ?10 vs HA ??10
• A random sample of size n10 is taken,
• and sample mean is calculated
• Accept H0 if
• Type I error?
• Type II error when ?10.10?

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Solution

• Type I errorP(reject H0 when ?10)

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Solution

• Type II errorP(accept H0 when H0 false)
• Power1-0.05710.9429

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12.3 12.4 Tests concerning Means

• A company claims its light bulbs last an average
  1,000 hours
• 5 steps to set up a statistical hypothesis test

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5 steps step 1

• 1. Set up H0 and HA
• H0 ?1,000 vs HA ?lt1,000
• This is a one-sided alternative.
• Other possibilities include
• H0 ?1,000 vs HA ??1,000
• (Two sided alternative)
• Note we could write H0 ? 1,000, but in this
  book H0 is always written with an equal () sign.

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5 steps step 2 and 3

• 2. Specify ?P(type I error) level of
  significance.
• ?0.05 usually. This corresponds to 95
  confidence.
• 3. Decide on sample size, n, and specify when to
  reject H0 based on some statistic so that
• ?P(Reject H0 when H0 is true)

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Step 3 continued

• Suppose we use n10 bulbs. Find the sample mean
  , and compare to 1000. We need to set a
  probability to a0.05, so we want a statistic we
  can compare to a table of probabilities.
• If we know ?, then set
• z has a standard normal distribution if ?1,000
  and then we can use the normal table.

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Step 3 continued

• Reject H0 ?1,000 in favor of HA ?lt1,000 if
  the sample mean is too far below 1000. This will
  give us a negative value of z. How far below 0
  does z have to be for us to reject H0?
• The rejection region is set up so that the
  probability of rejecting H0 is only a5 if H0 is
  true.
• So we reject H0 if

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Step 3 continued if s is unknown

• If ? is unknown, the usual situation, and the
  population is normal, we use a t distribution.
  Calculate sample deviation s
• Rejection region

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5 steps step 4

• 4. Collect the data and compute the statistics
  z or t
• Suppose , s30, n10 then

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5 steps step 5

• 5. Decide whether to reject H0
• t-3.16lt-1.833
• is in the rejection region
• Reject H0 ?1,000 in favor of HA ?lt1,000 at
  a0.05 level.

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5 steps summary

• 1. hypothesis statement
• 2. Specify level of significance a
• 3. determine the rejection region
• 4. compute the test statistic from data
• 5. conclusion

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Relationship Between Hypotheses Testing and
Confidence Intervals

• For two tailed test
• To accept null hypothesis at level ?
• H0 ??0
• is equivalent to showing ?0 is in the (1-?)
• Confidence Interval for ?.

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Example 12.3

• Normal population. ? unknown
• H0 ?750 vs HA ??750
• Define
• Reject H0 if the sample mean is too far from 750
  in either direction
• Rejection region

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Example

• Lets take ?0.05, n20 (df19)
• Data turned out to be
• Get t
• t0.0252.093
• tlt2.093
• Conclusion accept H0

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Example 12.3 (continued)

• ?50 is known, ?0.05, n20
• H0 ?750 vs HA ??750
• Reject region zgtz0.0251.96
• Calculate z
• zlt1.96. Accept H0
• Question if ?0.10, what is the conclusion?

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More cases

• H0 ?1000 vs HA ?gt1000
• Define t or z statistics
• (? unknown or known)
• Rejection regions
• tgtt? or zgtz?

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Rejection Regions
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P-value

• In practice more commonly one performs the test
  by computing a p-value.
• The book describes revised steps 3, 4, 5 as
• 3. We specify the test statistic.
• 4. Using the data we compute the test statistic
  and find its p-value.
• 5. If p-valuelta, reject H0.

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P-value

• The books definition of p-value A p-value is
  the lowest level of a at which we could reject
  H0.
• A more usual way to think about p-values
• If H0 is true, what is the probability of
  observing data with this much or more evidence
  against H0 in favor of HA?
• If H0 is true, what is the probability of
  observing data this far or farther from what we
  expect under H0 in favor of HA?
• If getting data this far or father from what we
  expect under H0 is small, reject H0.

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Example

• H0 m120 vs HA mlt120 a0.05
• Suppose from the data we get z-1.78
• Evidence against H0 is a sample mean quite a bit
  less than 120, meaning z quite a bit less than 0.
• If H0 is true, the probability of this much or
  more evidence against H0 in favor of HA is P(Z
  -1.78)
• P-valuelt0.05?Reject H0 m120 in favor of HA
  mlt120

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Example

• H0 m120 vs HA mlt120 a0.05
• Reject H0 at the a0.05 level since p-value lt
  0.05.
• The reject/accept H0 decision is the same as
  comparing z to -1.645, but the p-value gives more
  information
• How inconsistent are the data with H0?
• There is a 3.75 chance of seeing a mean at least
  this many SEs below 120 if in fact the true mean
  is 120.

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Example

• If we used a0.01, in order to reject H0 we would
  need to have the probability of this kind of data
  ( Z -1.78) to be 0.01 or less if H0 is true.
• Do not reject H0 at the a0.01 level
• p-value gt 0.01.
• There is more than a 1 chance of getting a
  sample mean 1.78 or more SEs below 120 if m120.
• Since this probability is not less than 1, dont
  reject H0.

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• If z is less than -1.645, then the p-value is
  less than 0.05 for HA mlt120.
• Comparing the p-value to 0.05 is the same as
  comparing the z value to -1.645.
• For t tests we cannot find the exact p-value
  without a calculator or software.
• On tests just give a range that the p-value falls
  into based on the table.
• Maybe 0.05 lt p-value lt 0.10

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P-value for 2-sided test

• H0 m120 vs HA m?120 a0.1
• Suppose from the data z1.32
• Evidence against H0 is z values away from 0 in
  either direction
• P-value20.09340.1868
• P-valuegta, Do not reject H0.
• HA mgt120, p-value 0.0934
• HA mlt120, p-value 1 0.09340.9066

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Exercise

• Given that n25, s100, and sample mean is 1050,
• 1. Test the hypotheses H0 m1000 vs HA mlt1000
  at level a0.05.
• 2. Test the hypotheses H0 m1000 vs HA m?1000
  at level a0.05.

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Solution

• 1.
• 2.

More evidence against H0 is smaller values of z
Evidence against H0 is z values away from 0 in
either direction
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A word of caution

• NOTE Accepting H0 does not prove H0 is true.
  There are many other possible values in the
  confidence interval.
• NOTE The p-value is NOT P(H0 is true)
• H0p0.5 HAp?0.5 p proby tack lands
  point up
• Toss n1 time. Tack lands point up.
• p-value 1.
• The data are perfectly consistent with H0
• We have absolutely no evidence against H0
• We cant say P(p0.5) 1.
• OPINION In most situations it is more useful to
  report confidence intervals rather than results
  of hypothesis tests.