MATH 200

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MATH 200 INTRODUCTION TO STATISTICS CHAPTER 8
Hypothesis Testing
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• Definition
• Hypothesis
• In statistics, a hypothesis is a claim or
  statement about a property of a population.

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Rare Event Rule for Inferential Statistics

• If, under a given assumption,
• the probability of an observed event
• is exceptionally small,
• we conclude that
• the assumption is probably not correct.

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Central Limit Theorem

• The Expected Distribution of Sample Means
  Assuming that ? 98.6

Likely sample means
or z - 6.64
Sample data x 98.20
µx 98.6
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Components of aFormal Hypothesis Test
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Null Hypothesis H0

• Statement about the value of a POPULATION
  PARAMETER
• Must contain condition of EQUALITY , , or
  
• Test the Null Hypothesis directly
• Reject H0 or fail to reject H0

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Alternative Hypothesis H1

• Must be true if H0 is false
• Must contain condition of INEQUALITY ? , lt ,
  or gt
• Opposite of Null Hypothesis

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Test Statistic

• A value computed from the sample data that is
  used in making the decision about whether to
  reject the null hypothesis

• For large samples, when testing claims about
  population means, the test statistic is a
  z-score corresponding to the sample mean.

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Critical Region

• Set of all values of the test statistic that
  would cause a rejection of the
• null hypothesis

Critical Region
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Critical Region

• Set of all values of the test statistic that
  would cause a rejection of the
• null hypothesis

Critical Region
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Critical Region

• Set of all values of the test statistic that
  would cause a rejection of the
• null hypothesis

Critical Regions
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Significance Level

• denoted by ?
• the probability that the test statistic will fall
  in the critical region when the null hypothesis
  is actually true.
• common choices are 0.05, 0.01, and 0.10

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Critical Value

• Value or values that separate the critical region
  (where we reject the null hypothesis) from the
  values of the test statistics that do not lead
  to a rejection of the null hypothesis

Fail to reject H0
Reject H0
Critical Value ( z score )
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Two-tailed, Right-tailed,Left-tailed Tests

• The tails in a distribution are the extreme
  regions bounded
• by critical values.

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Two-tailed Test

• H0 µ 100
• H1 µ ? 100

? is divided equally between the two tails of
the critical region
UNEQUAL means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100
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Right-tailed Test

• H0 µ ? 100
• H1 µ gt 100

Values that differ significantly from 100
100
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Left-tailed Test

• H0 µ ? 100
• H1 µ lt 100

Fail to reject H0
Reject H0
Values that differ significantly from 100
100
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Conclusions in Hypothesis Testing

• Always test the NULL hypothesis
• Reject H0
• Fail to reject H0
• Be careful to include the correct wording of the
  final conclusion

or
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Wording of Final Conclusion
Start
Only case in which original claim is rejected
Claim contains equality?
There is sufficient evidence to reject the
claim that. . . (original claim).
Yes
Yes Claim becomes H0
Reject H0?
No
There is not sufficient evidence to reject
the claim that (original claim).
No Claim becomes H1
Only case in which original claim is
supported
There is sufficient evidence to support the
claim that . . . (original claim).
Yes
Reject H0?
No
There is not sufficient evidence to support
the claim that (original claim).
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Fail to Reject versus Accept

• Some texts use accept the null hypothesis
• We are not proving the null hypothesis (cant
  PROVE equality)
• If the sample evidence is not strong enough to
  warrant rejection, then the null hypothesis may
  or may not be true (just as a defendant found NOT
  GUILTY may or may not be innocent)

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Type I Error

• Rejecting the null hypothesis when it is true.
• ???(alpha) represents the probability of a type
  I error
• Example Rejecting a claim that the mean body
  temperature is 98.6 degrees when the mean really
  is 98.6

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Type II Error

• Failing to reject the null hypothesis when it is
  false.
• ß (beta) represents the probability of a type
  II error
• Example Failing to reject the claim that the
  mean body temperature is 98.6 degrees when the
  mean really isnt 98.6

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Type I and Type II Errors
NULL HYPOTHESIS
TRUE
FALSE
Type I error a Rejecting a true null hypothesis
Reject the null hypothesis
CORRECT
DECISION
Type II error ß Failing to reject a false null
hypothesis
Fail to reject the null hypothesis
CORRECT
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Controlling Type I and Type II Errors

• ?, ?, and n are interrelated. If one is kept
  constant, then an increase in one of the
  remaining two will cause a decrease in the other.
  
• For any fixed ?, an increase in the sample size n
  will cause a ??????? in ??
• For any fixed sample size n , a decrease in ?
  will cause a ??????? in ?.
• Conversely, an increase in ? will cause a ???????
  in ? .
• To decrease both ? and ?, ??????? the sample
  size n.

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Hypothesis Testing Large Samples
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Assumptions

• for testing claims about population means
• The sample is a simple random sample.
• The sample is large (n gt 30).
• a) Central limit theorem applies
• b) Can use normal distribution
• If ? is unknown, we can use sample standard
  deviation s as estimate for ?.

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P-Value Methodof Testing Hypotheses

• Goal is to determine whether a sample result is
  significantly different from the claimed value
• Finds the probability (P-value) of getting a
  result and rejects the null hypothesis if that
  probability is very low

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Rare Event Rule for Inferential Statistics

• If, under a given assumption,
• the probability of an observed event
• is exceptionally small,
• we conclude that
• the assumption is probably not correct.

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P-Value Methodof Testing Hypotheses

• Definition
• P-Value (or probability value)
• The probability of getting a value of the sample
  test statistic that is at least as extreme as the
  one found from the sample data, assuming that the
  null hypothesis is true

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TRADITIONAL METHOD
critical region

• Use significance level to determine critical
  region.

Z0

• Calculate test statistic by converting sample
  mean to a z-score.

critical value

• Reject H0 if the test statistic falls in the
  critical region.

P-value area of this region
P-VALUE METHOD

• Calculate p-value of sample mean, assuming that
  H0 is true.

_ x
µ (according to H0)

• Reject H0 if the p-value is less than the
  significance level ?.

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P-value Method of Testing Hypotheses

• Write the CLAIM in symbolic form.

• Write H0 and H1. If the claim contains equality,
  it becomes H0. Otherwise it becomes H1. The other
  hypothesis is the symbolic form that must be true
  when the original claim is false.

• Select the significance level ? based on the
  seriousness of a type I error. Make ? small if
  the consequences of rejecting a true H0 are
  severe. Values of 0.05 and 0.01 are very common.
  Often the significance level will be given.

• Identify the appropriate distribution (normal
  distribution or t distribution).

• Find the P-value and draw a graph.

• Make a DECISION

• Reject the null hypothesis if the P-value is less
  than or equal to the significance level ?

• Fail to reject the null hypothesis if the P-value
  is greater than the significance level ?

• Write the CONCLUSION in simple non-technical
  terms.

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Wording of Final Conclusion
Start
Only case in which original claim is rejected
Claim contains equality?
There is sufficient evidence to reject the
claim that. . . (original claim).
Yes
Yes Claim becomes H0
Reject H0?
No
There is not sufficient evidence to reject
the claim that (original claim).
No Claim becomes H1
Only case in which original claim is
supported
There is sufficient evidence to support the
claim that . . . (original claim).
Yes
Reject H0?
No
There is not sufficient evidence to support
the claim that (original claim).
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Example The body temperatures of 106 healthy
adults were recorded. The mean was 98.2o and s
was 0.62o. At the 0.05 significance level, test
the claim that the mean body temperature of ALL
healthy adults is equal to 98.6o.

• CLAIM µ 98.6

• Graph and P-value

• H0 µ 98.6

H1 µ ? 98.6
? x 98.2
µ 98.6

• ? .05 (given in problem)

• Normal distribution (n gt 30)

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To Determine the P-value
STAT, TESTS, Z-Test Inpt Stats µ0 98.6
(population mean according to H0) s 0.62 (can
use s because n gt 30) x 98.2 (sample mean) n
106 (sample size) µ ? µ0 (according to
H1) CALCULATE (Can use DRAW to see the graph)
Calculator returns P 3.1039E-11, which is
equal to 0.000000000031039. Round to 0.
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Example The body temperatures of 106 healthy
adults were recorded. The mean was 98.2o and s
was 0.62o. At the 0.05 significance level, test
the claim that the mean body temperature of ALL
healthy adults is equal to 98.6o.

• CLAIM µ 98.6

• Graph and P-value

• H0 µ 98.6

H1 µ ? 98.6
? x 98.2
µ 98.6

• ? .05 (given in problem)

P 0

• Decision
• Reject H0 because P lt .05

• Normal distribution (n gt 30)

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Wording of Final Conclusion
Start
Only case in which original claim is rejected
Claim contains equality?
There is sufficient evidence to reject the
claim that. . . (original claim).
Yes
Yes Claim becomes H0
Reject H0?
No
There is not sufficient evidence to reject
the claim that (original claim).
No Claim becomes H1
Only case in which original claim is
supported
There is sufficient evidence to support the
claim that . . . (original claim).
Yes
Reject H0?
No
There is not sufficient evidence to support
the claim that (original claim).
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Example The body temperatures of 106 healthy
adults were recorded. The mean was 98.2o and s
was 0.62o. At the 0.05 significance level, test
the claim that the mean body temperature of ALL
healthy adults is equal to 98.6o.

• CLAIM µ 98.6

• Graph and P-value

• H0 µ 98.6

H1 µ ? 98.6
? x 98.2
µ 98.6

• ? .05 (given in problem)

P 0

• Decision
• Reject H0 because P lt .05

• Normal distribution (n gt 30)

• Conclusion There is sufficient evidence to
  reject the claim that the mean body temperature
  for healthy adults is 98.6.

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Rationale of Hypotheses Testing

• Based on RARE EVENT PRINCIPLE If, under a given
  assumption, the probability of getting an
  observed result is very small, we conclude that
  the assumption is probably not correct.
• When testing a claim, we make an assumption
  (null hypothesis) that contains equality. We
  compare the assumption and the sample results and
  form one of the following conclusions

• If the sample results can easily occur when the
  assumption (null hypothesis) is true, we
  attribute the relatively small discrepancy
  between the assumption and the sample results to
  chance.

• If the sample results cannot easily occur when
  the assumption (null hypothesis) is true, we
  explain the relatively large discrepancy between
  the assumption and the sample by concluding that
  the assumption is not true.

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Hypothesis Testing Small Samples
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Assumptions

• for testing claims about population means
• The sample is a simple random sample.
• The sample is small (n lt 30).
• The value of the population standard deviation ?
  is unknown.
• The sample values come from a population with a
  distribution that is approximately normal.

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Test Statistic for a Student t-distribution
x -µx
t
s
n

• Degrees of freedom (df) n -1

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Important Properties of the Student t
Distribution

• 1. The Student t distribution is different for
  different sample sizes (see Figure 6-5 in Section
  6-3).
• 2. The Student t distribution has the same
  general bell shape as the normal distribution
  its wider shape reflects the greater variability
  that is expected with small samples.
• 3. The Student t distribution has a mean of t 0
  (just as the standard normal distribution has a
  mean of z 0).
• 4. The standard deviation of the Student t
  distribution varies with the sample size and is
  greater than 1 (unlike the standard normal
  distribution, which has a ??? 1).
• 5. As the sample size n gets larger, the Student
  t distribution get closer to the normal
  distribution. For values of n gt 30, the
  differences are so small that we can use the
  critical z values instead of developing a much
  larger table of critical t values.

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Choosing between the Normal and Student
t-Distributionswhen Testing a Claim about a
Population Mean µ
Start
Use NORMAL distribution
Yes
n gt 30?
Use s if ? is unknown.
No
Is ? known ?
Use NORMAL distribution Extremely unusual!
Is ? known?
Population normally distributed?
Yes
Yes
No
No
Use nonparametric methods (not covered in this
course)
Use STUDENT t distribution
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EXAMPLE Chicken FeedUsing regular feed, a
poultry farmers newborn chickens have normally
distributed weights with a mean of 62.5 oz. With
enriched feed, the weights (in ounces) shown
below were observed 61.4 62.2 66.9 63.3 66.2 66.
0 63.1 63.7 66.6 Use a .05 significance level to
test the claim that the mean weight is higher
with the enriched feed.
5.

• Claim µ gt 62.5

P .013

• H0 µ 62.5
• H1 µ gt 62.5

? x
µ 62.5
P .013

• a 0.05

6. P lt .05 so reject H0
7. There is sufficient evidence to support the
claim that the mean weight is higher than 62.5
oz. with the enriched chicken feed.

• t distribution because n 30

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The larger Student t critical value shows that
with a small sample, the sample evidence must be
more extreme before we consider the difference is
significant.
EXAMPLE Given µ0 2, s 1, x 2.3, µ ? µ0
_
Use a Z-test to find the P-value with n 50.
Use a t-test to find the P-value with n 20.
P 0.0339
P 0.1955
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Hypothesis Testing Proportions
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Assumptions

• The sample is a simple random sample.
• There is a fixed number of independent trials,
  two categories of outcomes, and constant
  probabilities for each trial.
• The normal distribution can be used to
  approximate the distribution of sample
  proportions because np ? 5 and nq ? 5 are both
  satisfied.

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Notation
n number of trials
?
p x/n (sample proportion)

• p population proportion
• (used in the null hypothesis)
• q 1 - p

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Test Statistic for Testing a Claim about a
Proportion
?
p - p
z
pq
n
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P-value Method
Use the same procedure used on previous
hypothesis tests.

• Reject the null hypothesis if the P-value is less
  than or equal to the significance level ?.

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?
p sometimes is given directly 10 of the
observed sports cars are red is expressed
as p 0.10
?
?
p sometimes must be calculated 96 surveyed
households have cable TV
and 54 do not is calculated using
?
x
96
p 0.64
n
(9654)

• (determining the sample proportion of households
  with cable TV)

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Example A survey showed that among 785 randomly
selected factory workers, 23.7 smoke. Use a .01
level of significance to test the claim that the
rate of smoking among factory workers is less
than the 27 rate for the general population.
5. Use 1-PropZTest. .237 x 785 186.045 Use
186 for x. Graph not necessary for test of
proportion P 0.018

• Claim p lt .27

• H0 p .27
• H1 p lt .27

• a 0.01

• Fail to reject H0 because p gt .01

• Normal distribution because n gt 30

7. There is not sufficient evidence to support
the claim that the rate of smoking among factory
workers is less than the 27 rate for the general
population.