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Chapter 8 Statistical inference: Significance Tests About Hypotheses

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Title: Chapter 8 Statistical inference: Significance Tests About Hypotheses


1
Chapter 8Statistical inference Significance
Tests About Hypotheses
  • Learn .
  • To use an inferential method called
  • a Significance Test
  • To analyze evidence that data provide
  • To make decisions based on data

2
Two Major Methods for Making Statistical
Inferences about a Population
  • Confidence Interval
  • Significance Test

3
Questions that Significance Tests Attempt to
Answer
  • Does a proposed diet truly result in weight loss,
    on the average?
  • Is there evidence of discrimination against women
    in promotion decisions?
  • Does one advertising method result in better
    sales, on the average, than another advertising
    method?

4
Section 8.1
  • What Are the Steps For Performing a Significance
    Test?

5
Hypothesis
  • A hypothesis is a statement about a population,
    usually of the form that a certain parameter
    takes a particular numerical value or falls in a
    certain range of values
  • The main goal in many research studies is to
    check whether the data support certain hypotheses

6
Significance Test
  • A significance test is a method of using data to
    summarize the evidence about a hypothesis
  • A significance test about a hypothesis has five
    steps

7
Step 1 Assumptions
  • A (significance) test assumes that the data
    production used randomization
  • Other assumptions may include
  • Assumptions about the sample size
  • Assumptions about the shape of the population
    distribution

8
Step 2 Hypotheses
  • Each significance test has two hypotheses
  • The null hypothesis is a statement that the
    parameter takes a particular value
  • The alternative hypothesis states that the
    parameter falls in some alternative range of
    values

9
Null and Alternative Hypotheses
  • The value in the null hypothesis usually
    represents no effect
  • The symbol Ho denotes null hypothesis
  • The value in the alternative hypothesis usually
    represents an effect of some type
  • The symbol Ha denotes alternative hypothesis

10
Null and Alternative Hypotheses
  • A null hypothesis has a single parameter value,
    such as Ho p 1/3
  • An alternative hypothesis has a range of values
    that are alternatives to the one in Ho such as
  • Ha p ? 1/3 or
  • Ha p 1/3 or
  • Ha p

11
Step 3 Test Statistic
  • The parameter to which the hypotheses refer has a
    point estimate the sample statistic
  • A test statistic describes how far that estimate
    (the sample statistic) falls from the parameter
    value given in the null hypothesis

12
Step 4 P-value
  • To interpret a test statistic value, we use a
    probability summary of the evidence against the
    null hypothesis, Ho
  • First, we presume that Ho is true
  • Next, we consider the sampling distribution from
    which the test statistic comes
  • We summarize how far out in the tail of this
    sampling distribution the test statistic falls

13
Step 4 P-value
  • We summarize how far out in the tail the test
    statistic falls by the tail probability of that
    value and values even more extreme
  • This probability is called a P-value
  • The smaller the P-value, the stronger the
    evidence is against Ho

14
Step 4 P-value
15
Step 4 P-value
  • The P-value is the probability that the test
    statistic equals the observed value or a value
    even more extreme
  • It is calculated by presuming that the null
    hypothesis H is true

16
Step 5 Conclusion
  • The conclusion of a significance test reports the
    P-value and interprets what it says about the
    question that motivated the test

17
Summary The Five Steps of a Significance Test
  • Assumptions
  • Hypotheses
  • Test Statistic
  • P-value
  • Conclusion

18
Is the Statement a Null Hypothesis or an
Alternative Hypothesis?
  • In Canada, the proportion of adults who favor
    legalize gambling is 0.50.
  • Null Hypothesis
  • Alternative Hypothesis

19
Is the Statement a Null Hypothesis or an
Alternative Hypothesis?
  • The proportion of all Canadian college students
    who are regular smokers is less than 0.24, the
    value it was ten years ago.
  • Null Hypothesis
  • Alternative Hypothesis

20
Section 8.2
  • Significance Tests About
  • Proportions

21
Example Are Astrologers Predictions Better
Than Guessing?
  • Scientific test of astrology experiment
  • For each of 116 adult volunteers, an astrologer
    prepared a horoscope based on the positions of
    the planets and the moon at the moment of the
    persons birth
  • Each adult subject also filled out a California
    Personality Index Survey

22
Example Are Astrologers Predictions Better
Than Guessing?
  • For a given adult, his or her birth data and
    horoscope were shown to an astrologer together
    with the results of the personality survey for
    that adult and for two other adults randomly
    selected from the group
  • The astrologer was asked which personality chart
    of the 3 subjects was the correct one for that
    adult, based on his or her horoscope

23
Example Are Astrologers Predictions Better
Than Guessing?
  • 28 astrologers were randomly chosen to take part
    in the experiment
  • The National Council for Geocosmic Research
    claimed that the probability of a correct guess
    on any given trial in the experiment was larger
    than 1/3, the value for random guessing

24
Example Are Astrologers Predictions Better
Than Guessing?
  • Put this investigation in the context of a
    significance test by stating null and alternative
    hypotheses

25
Example Are Astrologers Predictions Better
Than Guessing?
  • With random guessing, p 1/3
  • The astrologers claim p 1/3
  • The hypotheses for this test
  • Ho p 1/3
  • Ha p 1/3

26
What Are the Steps of a Significance Test about a
Population Proportion?
  • Step 1 Assumptions
  • The variable is categorical
  • The data are obtained using randomization
  • The sample size is sufficiently large that the
    sampling distribution of the sample proportion is
    approximately normal
  • np 15 and n(1-p) 15

27
What Are the Steps of a Significance Test about a
Population Proportion?
  • Step 2 Hypotheses
  • The null hypothesis has the form
  • Ho p po
  • The alternative hypothesis has the form
  • Ha p po (one-sided test) or
  • Ha p
  • Ha p ? po (two-sided test)

28
What Are the Steps of a Significance Test about a
Population Proportion?
  • Step 3 Test Statistic
  • The test statistic measures how far the sample
    proportion falls from the null hypothesis value,
    po, relative to what wed expect if Ho were true
  • The test statistic is

29
What Are the Steps of a Significance Test about a
Population Proportion?
  • Step 4 P-value
  • The P-value summarizes the evidence
  • It describes how unusual the data would be if H0
    were true

30
What Are the Steps of a Significance Test about a
Population Proportion?
  • Step 5 Conclusion
  • We summarize the test by reporting and
    interpreting the P-value

31
Example Are Astrologers Predictions Better
Than Guessing?
  • Step 1 Assumptions
  • The data is categorical each prediction falls
    in the category correct or incorrect
    prediction
  • Each subject was identified by a random number.
    Subjects were randomly selected for each
    experiment.
  • np116(1/3) 15
  • n(1-p) 116(2/3) 15

32
Example Are Astrologers Predictions Better
Than Guessing?
  • Step 2 Hypotheses
  • H0 p 1/3
  • Ha p 1/3

33
Example Are Astrologers Predictions Better
Than Guessing?
  • Step 3 Test Statistic
  • In the actual experiment, the astrologers were
    correct with 40 of their 116 predictions (a
    success rate of 0.345)

34
Example Are Astrologers Predictions Better Than
Guessing?
  • Step 4 P-value
  • The P-value is 0.40

35
Example Are Astrologers Predictions Better Than
Guessing?
  • Step 5 Conclusion
  • The P-value of 0.40 is not especially small
  • It does not provide strong evidence against H0 p
    1/3
  • There is not strong evidence that astrologers
    have special predictive powers

36
How Do We Interpret the P-value?
  • A significance test analyzes the strength of the
    evidence against the null hypothesis
  • We start by presuming that H0 is true
  • The burden of proof is on Ha

37
How Do We Interpret the P-value?
  • The approach used in hypotheses testing is called
    a proof by contradiction
  • To convince ourselves that Ha is true, we must
    show that data contradict H0
  • If the P-value is small, the data contradict H0
    and support Ha

38
Two-Sided Significance Tests
  • A two-sided alternative hypothesis has the form
    Ha p ? p0
  • The P-value is the two-tail probability under the
    standard normal curve
  • We calculate this by finding the tail probability
    in a single tail and then doubling it

39
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Study investigate whether dogs can be trained
    to distinguish a patient with bladder cancer by
    smelling compounds released in the patients urine

40
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Experiment
  • Each of 6 dogs was tested with 9 trials
  • In each trial, one urine sample from a bladder
    cancer patient was randomly place among 6 control
    urine samples

41
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Results
  • In a total of 54 trials with the six dogs, the
    dogs made the correct selection 22 times (a
    success rate of 0.407)

42
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Does this study provide strong evidence that the
    dogs predictions were better or worse than with
    random guessing?

43
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Step 1 Check the sample size requirement
  • Is the sample size sufficiently large to use the
    hypothesis test for a population proportion?
  • Is np0 15 and n(1-p0) 15?
  • 54(1/7) 7.7 and 54(6/7) 46.3
  • The first, np0 is not large enough
  • We will see that the two-sided test is robust
    when this assumption is not satisfied

44
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Step 2 Hypotheses
  • H0 p 1/7
  • Ha p ? 1/7

45
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Step 3 Test Statistic

46
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Step 4 P-value

47
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Step 5 Conclusion
  • Since the P-value is very small and the sample
    proportion is greater than 1/7, the evidence
    strongly suggests that the dogs selections are
    better than random guessing

48
Example Dr Dog Can Dogs Detect Cancer by
Smell?
  • Insight
  • In this study, the subjects were a convenience
    sample rather than a random sample from some
    population
  • Also, the dogs were not randomly selected
  • Any inferential predictions are highly tentative
  • The predictions become more conclusive if similar
    results occur in other studies

49
Summary of P-values for Different Alternative
Hypotheses
50
The Significance Level Tells Us How Strong the
Evidence Must Be
  • Sometimes we need to make a decision about
    whether the data provide sufficient evidence to
    reject H0
  • Before seeing the data, we decide how small the
    P-value would need to be to reject H0
  • This cutoff point is called the significance
    level

51
The Significance Level Tells Us How Strong the
Evidence Must Be
52
Significance Level
  • The significance level is a number such that we
    reject H0 if the P-value is less than or equal to
    that number
  • In practice, the most common significance level
    is 0.05
  • When we reject H0 we say the results are
    statistically significant

53
Possible Decisions in a Test with Significance
Level 0.05
54
Report the P-value
  • Learning the actual P-value is more informative
    than learning only whether the test is
    statistically significant at the 0.05 level
  • The P-values of 0.01 and 0.049 are both
    statistically significant in this sense, but the
    first P-value provides much stronger evidence
    against H0 than the second

55
Do Not Reject H0 Is Not the Same as Saying
Accept H0
  • Analogy Legal trial
  • Null Hypothesis Defendant is Innocent
  • Alternative Hypothesis Defendant is Guilty
  • If the jury acquits the defendant, this does not
    mean that it accepts the defendants claim of
    innocence
  • Innocence is plausible, because guilt has not
    been established beyond a reasonable doubt

56
One-Sided vs Two-Sided Tests
  • Things to consider in deciding on the alternative
    hypothesis
  • The context of the real problem
  • In most research articles, significance tests use
    two-sided P-values
  • Confidence intervals are two-sided

57
The Binomial Test for Small Samples
  • The test about a proportion assumes normal
    sampling distributions for and the z-test
    statistic.
  • It is a large-sample test the requires that the
    expected numbers of successes and failures be at
    least 15. In practice, the large-sample z test
    still performs quite well in two-sided
    alternatives even for small samples.
  • Warning For one-sided tests, when p0 differs
    from 0.50, the large-sample test does not work
    well for small samples

58
For a test of H0 p 0.50
  • The z test statistic is 1.04. Find the
    P-value for Ha p 0.50.
  • .15
  • .20
  • .175
  • .222

59
For a test of H0 p 0.50
  • The z test statistic is 1.04. Find the
    P-value for Ha p ? 0.50.
  • .15
  • .22
  • .30
  • .175

60
For a test of H0 p 0.50
  • The z test statistic is 1.04. Does the
    P-value for Ha p ? 0.50 give strong evidence
    against H0?
  • yes
  • no

61
For a test of H0 p 0.50
  • The z test statistic is 2.50. Find the
    P-value for Ha p 0.50.
  • .05
  • .10
  • .0062
  • .0124

62
For a test of H0 p 0.50
  • The z test statistic is 2.50. Find the
    P-value for Ha p ? 0.50.
  • .05
  • .10
  • .0062
  • .0124

63
For a test of H0 p 0.50
  • The z test statistic is 2.50. Does the
    P-value for Ha p ? 0.50 give strong evidence
    against H0?
  • yes
  • no

64
Section 8.3
  • Significance Tests about Means

65
What Are the Steps of a Significance Test about a
Population Mean?
  • Step 1 Assumptions
  • The variable is quantitative
  • The data are obtained using randomization
  • The population distribution is approximately
    normal. This is most crucial when n is small and
    Ha is one-sided.

66
What Are the Steps of a Significance Test about a
Population Mean?
  • Step 2 Hypotheses
  • The null hypothesis has the form
  • H0 µ µ0
  • The alternative hypothesis has the form
  • Ha µ µ0 (one-sided test) or
  • Ha µ
  • Ha µ ? µ0 (two-sided test)

67
What Are the Steps of a Significance Test about a
Population Mean?
  • Step 3 Test Statistic
  • The test statistic measures how far the sample
    mean falls from the null hypothesis value µ0
    relative to what wed expect if H0 were true
  • The test statistic is

68
What Are the Steps of a Significance Test about a
Population Mean?
  • Step 4 P-value
  • The P-value summarizes the evidence
  • It describes how unusual the data would be if H0
    were true

69
What Are the Steps of a Significance Test about a
Population Mean?
  • Step 5 Conclusion
  • We summarize the test by reporting and
    interpreting the P-value

70
Summary of P-values for Different Alternative
Hypotheses
71
Example Mean Weight Change in Anorexic Girls
  • A study compared different psychological
    therapies for teenage girls suffering from
    anorexia
  • The variable of interest was each girls weight
    change weight at the end of the study
    weight at the beginning of the study

72
Example Mean Weight Change in Anorexic Girls
  • One of the therapies was cognitive therapy
  • In this study, 29 girls received the therapeutic
    treatment
  • The weight changes for the 29 girls had a sample
    mean of 3.00 pounds and standard deviation of
    7.32 pounds

73
Example Mean Weight Change in Anorexic Girls
74
Example Mean Weight Change in Anorexic Girls
  • How can we frame this investigation in the
    context of a significance test that can detect a
    positive or negative effect of the therapy?
  • Null hypothesis no effect
  • Alternative hypothesis therapy has some
    effect

75
Example Mean Weight Change in Anorexic Girls
  • Step 1 Assumptions
  • The variable (weight change) is quantitative
  • The subjects were a convenience sample, rather
    than a random sample. The question is whether
    these girls are a good representation of all
    girls with anorexia.
  • The population distribution is approximately
    normal

76
Example Mean Weight Change in Anorexic Girls
  • Step 2 Hypotheses
  • H0 µ 0
  • Ha µ ? 0

77
Example Mean Weight Change in Anorexic Girls
  • Step 3 Test Statistic

78
Example Mean Weight Change in Anorexic Girls
  • Step 4 P-value
  • Minitab Output
  • Test of mu 0 vs not 0
  • Variable N Mean StDev SE Mean
    wt_chg 29 3.000 7.3204 1.3594 CI
  • 95 CI T P
  • (0.21546, 5.78454) 2.21 0.036

79
Example Mean Weight Change in Anorexic Girls
  • Step 5 Conclusion
  • The small P-value of 0.036 provides considerable
    evidence against the null hypothesis (the
    hypothesis that the therapy had no effect)

80
Example Mean Weight Change in Anorexic Girls
  • The diet had a statistically significant
    positive effect on weight (mean change 3
    pounds, n 29, t 2.21, P-value 0.04)
  • The effect, however, may be small in practical
    terms
  • 95 CI for µ (0.2, 5.8) pounds

81
Results of Two-Sided Tests and Results of
Confidence Intervals Agree
  • Conclusions about means using two-sided
    significance tests are consistent with
    conclusions using confidence intervals
  • If P-value 0.05 in a two-sided test, a 95
    confidence interval does not contain the H0
    value
  • If P-value 0.05 in a two-sided test, a 95
    confidence interval does contain the H0 value

82
What If the Population Does Not Satisfy the
Normality Assumption
  • For large samples (roughly about 30 or more) this
    assumption is usually not important
  • The sampling distribution of x is approximately
    normal regardless of the population distribution

83
What If the Population Does Not Satisfy the
Normality Assumption
  • In the case of small samples, we cannot assume
    that the sampling distribution of x is
    approximately normal
  • Two-sided inferences using the t distribution are
    robust against violations of the normal
    population assumption
  • They still usually work well if the actual
    population distribution is not normal

84
Regardless of Robustness, Look at the Data
  • Whether n is small or large, you should look at
    the data to check for severe skew or for severe
    outliers
  • In these cases, the sample mean could be a
    misleading measure

85
A study has a random sample of 20 subjects. The
test statistic for testing Hoµ100 is t 2.40.
  • Find the approximate P-value for the alternative,
    Ha µ 100.
  • between .100 and .050
  • between .050 and .025
  • between .025 and .010
  • between .010 and .005

86
A study has a random sample of 20 subjects. The
test statistic for testing Hoµ100 is t 2.40.
  • Find the approximate P-value for the alternative,
    Ha µ ? 100.
  • between .100 and .050
  • between .050 and .020
  • between .025 and .010
  • between .020 and .010

87
Section 8.4
  • Decisions and Types of Errors in Significance
    Tests

88
Type I and Type II Errors
  • When H0 is true, a Type I Error occurs when H0 is
    rejected
  • When H0 is false, a Type II Error occurs when H0
    is not rejected

89
Significance Test Results
90
An Analogy Decision Errors in a Legal Trial
91
P(Type I Error) Significance Level a
  • Suppose H0 is true. The probability of rejecting
    H0, thereby committing a Type I error, equals the
    significance level, a, for the test.

92
P(Type I Error)
  • We can control the probability of a Type I error
    by our choice of the significance level
  • The more serious the consequences of a Type I
    error, the smaller a should be

93
Type I and Type II Errors
  • As P(Type I Error) goes Down, P(Type II Error)
    goes Up
  • The two probabilities are inversely related

94
A significance test about a proportion is
conducted using a significance level of 0.05.
  • The test statistic is 2.58. The P-value is 0.01.
    If Ho is true, for what probability of a Type I
    error was the test designed?
  • .01
  • .05
  • 2.58
  • .02

95
A significance test about a proportion is
conducted using a significance level of 0.05.
  • The test statistic is 2.58. The P-value is 0.01.
    If this test resulted in a decision error, what
    type of error was it?
  • Type I
  • Type II

96
Section 8.5
  • Limitations of Significance Tests

97
Statistical Significance Does Not Mean Practical
Significance
  • When we conduct a significance test, its main
    relevance is studying whether the true parameter
    value is
  • Above, or below, the value in H0 and
  • Sufficiently different from the value in H0 to be
    of practical importance

98
What the Significance Test Tells Us
  • The test gives us information about whether the
    parameter differs from the H0 value and its
    direction from that value

99
What the Significance Test Does Not Tell Us
  • It does not tell us about the practical
    importance of the results

100
Statistical Significance vs. Practical
Significance
  • A small P-value, such as 0.001, is highly
    statistically significant, but it does not imply
    an important finding in any practical sense
  • In particular, whenever the sample size is large,
    small P-values can occur when the point estimate
    is near the parameter value in H0

101
Significance Tests Are Less Useful Than
Confidence Intervals
  • A significance test merely indicates whether the
    particular parameter value in H0 is plausible
  • When a P-value is small, the significance test
    indicates that the hypothesized value is not
    plausible, but it tells us little about which
    potential parameter values are plausible

102
Significance Tests are Less Useful than
Confidence Intervals
  • A Confidence Interval is more informative,
    because it displays the entire set of believable
    values

103
Misinterpretations of Results of Significance
Tests
  • Do Not Reject H0 does not mean Accept H0
  • A P-value above 0.05 when the significance level
    is 0.05, does not mean that H0 is correct
  • A test merely indicates whether a particular
    parameter value is plausible

104
Misinterpretations of Results of Significance
Tests
  • Statistical significance does not mean practical
    significance
  • A small P-value does not tell us whether the
    parameter value differs by much in practical
    terms from the value in H0

105
Misinterpretations of Results of Significance
Tests
  • The P-value cannot be interpreted as the
    probability that H0 is true

106
Misinterpretations of Results of Significance
Tests
  • It is misleading to report results only if they
    are statistically significant

107
Misinterpretations of Results of Significance
Tests
  • Some tests may be statistically significant just
    by chance

108
Misinterpretations of Results of Significance
Tests
  • True effects may not be as large as initial
    estimates reported by the media

109
Section 8.6
  • How Likely is a Type II Error?

110
Type II Error
  • A Type II error occurs in a hypothesis test when
    we fail to reject H0 even though it is actually
    false

111
Calculating the Probability of a Type II Error
  • To calculate the probability of a Type II error,
    we must do a separate calculation for various
    values of the parameter of interest

112
Example Reconsider the Experiment to test
Astrologers Predictions
  • Scientific test of astrology experiment
  • For each of 116 adult volunteers, an astrologer
    prepared a horoscope based on the positions of
    the planets and the moon at the moment of the
    persons birth
  • Each adult subject also filled out a California
    Personality Index Survey

113
Example Reconsider the Experiment to test
Astrologers Predictions
  • For a given adult, his or her birth data and
    horoscope were shown to an astrologer together
    with the results of the personality survey for
    that adult and for two other adults randomly
    selected from the group
  • The astrologer was asked which personality chart
    of the 3 subjects was the correct one for that
    adult, based on his or her horoscope

114
Example Reconsider the Experiment to test
Astrologers Predictions
  • 28 astrologers were randomly chosen to take part
    in the experiment
  • The National Council for Geocosmic Research
    claimed that the probability of a correct guess
    on any given trial in the experiment was larger
    than 1/3, the value for random guessing

115
Example Reconsider the Experiment to test
Astrologers Predictions
  • With random guessing, p 1/3
  • The astrologers claim p 1/3
  • The hypotheses for this test
  • Ho p 1/3
  • Ha p 1/3
  • The significance level used for the test is 0.05

116
Example Reconsider the Experiment to test
Astrologers Predictions
  • For what values of the sample proportion can we
    reject H0?
  • A test statistic of z 1.645 has a P-value of
    0.05. So, we reject H0 for z 1.645 and we fail
    to reject H0 for z

117
Example Reconsider the Experiment to test
Astrologers Predictions
  • Find the value of the sample proportion that
    would give us a z of 1.645

118
Example Reconsider the Experiment to test
Astrologers Predictions
  • So, we fail to reject H0 if

  • Suppose that in reality astrologers can make the
    correct prediction 50 of the time (that is, p
    0.50)
  • In this case, (p 0.50), we can now calculate
    the probability of a Type II error

119
Example Reconsider the Experiment to test
Astrologers Predictions
  • We calculate the probability of a sample
    proportion proportion is 0.50

120
Example Reconsider the Experiment to test
Astrologers Predictions
  • The area to the left of -2.04 in the standard
    normal table is 0.02
  • The probability of making a Type II error and
    failing to reject H0 p 1/3 is only 0.02 in the
    case in which the true proportion is 0.50
  • This is only a small chance of making a Type II
    error

121
Power of a Test
  • Power 1 P(Type II error)
  • The higher the power, the better
  • In practice, it is ideal for studies to have high
    power while using a relatively small significance
    level

122
Example Reconsider the Experiment to test
Astrologers Predictions
  • In this example, the Power of the test at p
    0.50 is 1 0.02 0.98
  • Since, the higher the power the better, a test
    power of 0.98 is quite good
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