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Title: Introduction to statistics


1
Introduction to statistics
  • Programme Bioinformatics
  • Master Grid Computing
  • April 2007

Prof dr AHC van Kampen Bioinformatics
Laboratory KEBB, AMC
2
Descriptive Statistics
3
Describing data
4
Quartile
  • In descriptive statistics, a quartile is any of
    the three values which divide the sorted data set
    into four equal parts, so that each part
    represents 1/4th of the sample or population.
  • first quartile (designated Q1) lower quartile
    cuts off lowest 25 of data 25th percentile
  • second quartile (designated Q2) median cuts
    data set in half 50th percentile
  • third quartile (designated Q3) upper quartile
    cuts off highest 25 of data, or lowest 75
    75th percentile
  • The difference between the upper and lower
    quartiles is called the interquartile range.

5
Variance, S.D. of a Sample
variance
Degrees of freedom
Standard deviation
In statistics, the term degrees of freedom (df)
is a measure of the number of independent pieces
of information on which the precision of a
parameter estimate is based
6
Skewness
7
Box-whisker plots
8
Distributions
  • Normal, binomial, Poisson, hypergeometric,
    t-distribution, chi-square
  • What parameters describe their shapes
  • How these distributions can be useful

9
Normal distribution
10
The Normal Distribution
  • Also called a Gaussian distribution
  • Centered around the mean ? with a width
    determined by the standard deviation ?
  • Total area under the curve 1.0

11
A Normal Distribution . . .
  • For a mean of 5 and a standard deviation of 1

12
What Does a Normal Distribution Describe?
  • Imagine that you go to the lab and very carefully
    measure out 5 ml of liquid and weigh it.
  • Imagine repeating this process many times.
  • You wont get the same answer every time, but if
    you make a lot of measurements, a histogram of
    your measurements will approach the appearance of
    a normal distribution.

13
What Does a Normal Distribution Describe?
  • Any situation in which the exact value of a
    continuous variable is altered randomly from
    trial to trial.
  • The random uncertainty or random error

14
How Do You Use The Normal Distribution?
  • Use the area UNDER the normal distribution
  • For example, the area under the curve between xa
    and xb is the probability that your next
    measurement of x will fall between a and b

15
A normal distribution with a mean of 75 and a
standard deviation of 10. The shaded area
contains 95 of the area and extends from 55.4 to
94.6. For all normal distributions, 95 of the
area is within 1.96 standard deviations of the
mean.
16
How Do You Get ? and ??
  • To draw a normal distribution you must know ? and
    ?
  • If you made an infinite number of measurements,
    their mean would be ? and their standard
    deviation would be ?
  • In practice, you have a finite number of
    measurements with mean x and standard deviation s
  • For now, ? and ? will be given
  • Later well use x and s to estimate ? and ?

17
The Standard Normal Distribution
  • It is tedious to integrate a new normal
    distribution for every population, so use a
    standard normal distribution with standard
    tabulated areas.
  • Convert your measurement x to a standard score
    (z-score)
  • z (x - ?) / ?
  • Use the standard normal distribution
  • ? 0 and ? 1
  • (areas tabulated in any statistics text book)

The z-score indicates the number of standard
deviations that value x is away from the mean ?
18
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19
Probability density function
z-transform
green curve is standard normal distribution
20
Cummulative distribution functions
21
Exercises 1
If scores are normally distributed with a mean
of 30 and a standard deviation of 5, what
percent of the scores is (a) greater than 30?
(b) greater than 37? (c) between 28 and
34? What proportion of a normal distribution is
within one standard deviation of the mean?
What proportion is more than 1.8 standard
deviations from the mean? A test is normally
distributed with a mean of 40 and a standard
deviation of 7. What value would be needed to be
in the 85th percentile?
Stat tables http//www.statsoft.com/textbook/stta
ble.html
22
Binomial distribution
23
What Does the Binomial Distribution Describe?
  • yes/no experiments (two possible outcomes)
  • The probability of getting all tails if you
    throw a coin three times
  • The probability of getting all male puppies in a
    litter of 8
  • The probability of getting two defective
    batteries in a package of six

24
Exercise 2
  • What is the probability of getting one 2 when
    you roll six dice?

25
The Binomial Distribution
  • The probability of getting the result of interest
    k times out of n, if the overall probability of
    the result is p
  • Note that here, k is a discrete variable
  • Integer values only

bionomial coefficient
26
Binomial Distribution
  • n 6 number of dice rolled
  • p 1/6 probability of rolling a 2
  • k 0 1 2 3 4 5 6 of 2s out of 6

0.402
27
Binomial Distribution
  • n 8 number of puppies in litter
  • p 1/2 probability of any pup being male
  • k 0 1 2 3 4 5 6 7 8 of males out of 8

28
The Shape of the Binomial Distribution
  • Shape is determined by values of n and p
  • Only truly symmetric if p 0.5
  • Approaches normal distribution if n is large,
    unless ? is very small
  • Mean number of successes is np
  • Variance of distribution is
  • variance (X) n p(1- p)

29
Exercise 3
  • While you are in the bathroom, your little
    brother claims to have rolled a Yahtzee in 6s
    (five dice all 6s) in one roll of the five dice.
    How justified would you be in beating him up for
    cheating?

30
Poisson distribution
P(µ) probability of getting n counts (0, 1,
2,...) µ average of distribution
variance mean
31
Poisson distribution
Randomly placed dots over 50 scale divisions. On
average µ1 dot per interval
µ1
P(µ) probability of getting n counts µ average
of distribution
n
32
Exercise 4
Pn(µ) probability of getting n counts µ average
of distribution
Average number of phone calls in 1 hour
2.1 What is probability of getting 4 calls?
33
Exercise 5
Pn(µ) probability of getting a discrete value
n µ average of distribution
Average number of phone calls in 1 hour
2.1 What is probability of getting 0
calls? Does this simply the formula?
34
Hypergeometric distribution
35
Hypergeometric Distribution
  • Suppose that we have an urn with N balls in it
    of these m are white and others are black.
  • Then k balls are drawn from the urn without
    replacement and of these X are observed to be
    white.
  • X is a random variable following hypergeometric
    distribution

N20 mn10
draw k10 balls
X6
36
Hypergeometric Distribution
P(X x)

37
Fishers Exact Test
  • We often want to ask whether there are more white
    balls in the sample than expected by chance.

P(X ? x)
  • If the probability is small, it is less likely
    that we get the result by chance.

38
Hypergeometric example
  • Extract a cluster of 36 samples from leukemia
    microarray dataset
  • Whole dataset 47 ALL 25 AML
  • Extracted 29 ALL 7 AML
  • Is this sample enriched for ALL samples?

Pr(extracted ALL 29)

0.006
  • Conclusion This cluster is significantly
    enriched with ALL samples.

39
Sampling Distribution
  • Every time we take a random sample and calculate
    a statistic, the value of the statistic changes
    (remember, a statistic is a random variable).
  • If we continue to take random samples and
    calculate a given statistic over time, we will
    build up a distribution of values for the
    statistic. This distribution is referred to as a
    sampling distribution.
  • A sampling distribution is a distribution that
    describes the chance fluctuations of a statistic
    calculated from a random sample.

40
Sampling Distribution of the Mean
  • The probability distribution of is called
    the sampling distribution of the mean.
  • The distribution of , for a given sample
    size, n, describes the variability of sample
    averages around the population mean µ.

41
Sampling Distribution of the Mean
  • If a random sample of size n is taken from a
    normal population having mean µx and variance
    , then is a random variable which is also
    normally distributed with mean µx and variance
    .
  • Further,
  • is a standard normal random variable.

42
Sampling Distribution of the Mean
Original population
1
3
n(100,5)
n(100,1.58)
2
4
n(100,3.54)
n(100,1)
5/sqrt(2)3.54
43
Sampling Distribution of the Mean
  • Example A manufacturer of steel rods claims that
    the length of his bars follows a normal
    distribution with a mean of 30 cm and a standard
    deviation of 0.5 cm.
  • Assuming that the claim is true, what is the
    probability that a given bar will exceed 30.1 cm?
  • (b) Assuming the claim is true, what is the
    probability that the mean of 10 randomly chosen
    bars will exceed 30.1 cm?
  • (c) Assuming the claim is true, what is the
    probability that the mean of 100 randomly chosen
    bars will exceed 30.1 cm?

44
Sampling Distribution of the Mean
  • Example A manufacturer of steel rods claims that
    the length of his bars follows a normal
    distribution with a mean of 30 cm and a standard
    deviation of 0.5 cm.
  • Assuming that the claim is true, what is the
    probability that a given bar will exceed 30.1 cm?
    (z30.1-30)/0.50.2 ?p0.42)
  • (b) Assuming the claim is true, what is the
    probability that the mean of 10 randomly chosen
    bars will exceed 30.1 cm?
  • (z30.1-30)/(0.5/sqrt(10)0.63 ?p0.26)
  • (c) Assuming the claim is true, what is the
    probability that the mean of 100 randomly chosen
    bars will exceed 30.1 cm?
  • (z30.1-30)/(0.5/sqrt(100)2 ?p0.02)

45
Sampling Distribution of the Mean
.42
.42
. 26
.02
46
Inference on Population Mean
  • Example Suppose that it is very important to our
    manufacturing process that we detect a deviation
    in the bar mean of 0.1 cm or more.
  • Will sampling one bar allow us to detect a shift
    of 0.1 cm in the population mean?
  • Will sampling ten bars allow us to detect a
    shift of 0.1 cm in the population mean?
  • Will sampling one hundred bars allow us to
    detect a shift of 0.1 cm in the population mean?

47
Inference on Population Mean
48
Inference on Population Mean
49
Inference on Population Mean
50
Properties of Sample Mean as Estimator of
Population Mean
  • Expected value of sample mean is population mean
  • Among UNBIASED estimators, the mean has the
    SMALLEST variance
  • Variance

UNBIASED
??
??
_

??
_
As n increase, decrease.
standard error
x
x
51
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52
When the Population is Normal Sampling
Distribution is Also Normal
Population Distribution
Central Tendency
??
??

_
x
Variation
??
Sampling Distributions
??
_

x
n 16??X 2.5
n 4??X 5
53
Central Limit Theorem
As Sample Size Gets Large Enough
Sampling Distribution
Becomes almost normal regardless of shape of
population
54
When The Population is Not Normal
Population Distribution
Central Tendency
? 10
??
??

_
x
Variation
? 50
X
??
Sampling Distributions
??
_

x
n 30??X 1.8
n 4??X 5
55
Central Limit Theorem
  • As the sample size increases the sampling
    distribution of the sample mean approaches the
    normal distribution with mean ? and variance
    ?2/n

56
Example Sampling Distribution
Standardized
Sampling Distribution
Normal Distribution

? 1
.3830
.1915
.1915
Z
???? 0
7.8 8 8.2
57
Central Limit Theorem
  • Rule of thumb normal approximation for will
    be good if n gt 30. If n lt 30, the approximation
    is only good if the population from which you are
    sampling is not too different from normal.
  • Otherwise t-distribution

58
t-Distribution
  • So far, we have been assuming that we knew the
    value of s. This may be true if one has a large
    amount of experience with a certain process.
  • However, it is often true that one is estimating
    s along with µ from the same set of data.

59
t-Distribution
  • To allow for such a situation, we will consider
    the t statistic
  • which follows a t-distribution.

? standard error of the mean
60
t-Distribution
t(n?) Z
t(n6)
t(n3)
61
t-Distribution
  • If is the mean of a random sample of size n
    taken from a normal population having the mean µ
    and variance s2, and
  • then
  • is a random variable following the t-
    distribution with parameter ? n 1, where ? is
    degrees of freedom.

62
t-Distribution
  • The t-distribution has been tabularized.
  • ta represents the t-value that has an area of a
    to the right of it.
  • Note, due to symmetry, t1-a -ta

t.05
t.80
t.20
t.95
63
Example t-Distribution
  • The resistivity of batches of electrolyte follow
    a normal distribution. We sample 5 batches and
    get the following readings 1400, 1450, 1375,
    1500, 1550.
  • Does this data support or refute a population
    average of 1400?

64
Example t-Distribution
Support
p0.025
Refute
Refute
t2.78
1.71
65
Sampling Distribution of the Variance
  • The probability distribution of S2 is called the
    sampling distribution of the Variance.
  • The distribution of S2, for a given sample size,
    n, describes the variability of sample variances
    around the population variance s2.

66
Sampling Distribution S2
  • If S2 is the variance of a random sample of size
    n taken from a normal population having the
    variance s2, then the statistic
  • has a chi-squared distribution with ? n 1
    degrees of freedom.

67
Chi-Squared Distribution
?2(n3)
?2(n6)
?2(n11)
68
  • Introduction to Hypothesis Testing

69
Nonstatistical Hypothesis Testing
  • A criminal trial is an example of hypothesis
    testing without the statistics.
  • In a trial a jury must decide between two
    hypotheses. The null hypothesis is
  • H0 The defendant is innocent
  • The alternative hypothesis or research hypothesis
    is
  • H1 The defendant is guilty
  • The jury does not know which hypothesis is true.
    They must make a decision on the basis of
    evidence presented.

70
Nonstatistical Hypothesis Testing
  • In the language of statistics convicting the
    defendant is called rejecting the null hypothesis
    in favor of the alternative hypothesis. That is,
    the jury is saying that there is enough evidence
    to conclude that the defendant is guilty (i.e.,
    there is enough evidence to support the
    alternative hypothesis).
  • If the jury acquits it is stating that there is
    not enough evidence to support the alternative
    hypothesis. Notice that the jury is not saying
    that the defendant is innocent, only that there
    is not enough evidence to support the alternative
    hypothesis. That is why we never say that we
    accept the null hypothesis.

71
Nonstatistical Hypothesis Testing
  • There are two possible errors.
  • A Type I error occurs when we reject a true null
    hypothesis. That is, a Type I error occurs when
    the jury convicts an innocent person.
  • A Type II error occurs when we dont reject a
    false null hypothesis. That occurs when a guilty
    defendant is acquitted.

72
Nonstatistical Hypothesis Testing
  • The probability of a Type I error is denoted as
    a.
  • The probability of a type II error is ß.
  • The two probabilities are inversely related.
    Decreasing one increases the other.

73
Nonstatistical Hypothesis Testing
  • In the (US) system Type I errors are regarded as
    more serious. We try to avoid convicting innocent
    people. We are more willing to acquit guilty
    people.
  • We arrange to make a small by requiring the
    prosecution to prove its case and instructing the
    jury to find the defendant guilty only if there
    is evidence beyond a reasonable doubt.

74
Nonstatistical Hypothesis Testing
  • The critical concepts are these
  • There are two hypotheses, the null and the
    alternative hypotheses.
  • 2. The procedure begins with the assumption that
    the null hypothesis is true.
  • 3. The goal is to determine whether there is
    enough evidence to infer that the alternative
    hypothesis is true.
  • 4. There are two possible decisions
  • Conclude that there is enough evidence to support
    the alternative hypothesis.
  • Conclude that there is not enough evidence to
    support the alternative hypothesis.

75
Nonstatistical Hypothesis Testing
  • 5. Two possible errors can be made.
  • Type I error Reject a true null hypothesis
  • Type II error Do not reject a false null
    hypothesis.
  • P(Type I error) a
  • P(Type II error) ß

76
Introduction
  • Hypothesis testing is a procedure for making
    inferences about a population.
  • Hypothesis testing allows us to determine whether
    enough statistical evidence exists to conclude
    that a belief (i.e. hypothesis) about a parameter
    is supported by the data.

77
Concepts of Hypothesis Testing (1)
  • There are two hypotheses. One is called the null
    hypothesis and the other the alternative or
    research hypothesis. The usual notation is
  • H0 the null hypothesis
  • H1 the alternative or research hypothesis
  • The null hypothesis (H0) will always state that
    the parameter equals the value specified in the
    alternative hypothesis (H1)

78
Concepts of Hypothesis Testing
  • Consider example mean demand for computers
    during assembly lead time. Rather than estimate
    the mean demand, our operations manager wants to
    know whether the mean is different from 350
    units. We can rephrase this request into a test
    of the hypothesis
  • H0 350
  • Thus, our research hypothesis becomes
  • H1 ? 350

This is what we are interested in determining
79
Concepts of Hypothesis Testing
  • The testing procedure begins with the assumption
    that the null hypothesis is true.
  • Thus, until we have further statistical evidence,
    we will assume
  • H0 350 (assumed to be TRUE)

80
Concepts of Hypothesis Testing
  • The goal of the process is to determine whether
    there is enough evidence to infer that the
    alternative hypothesis is true.
  • That is, is there sufficient statistical
    information to determine if this statement
  • H1 ? 350, is true?

This is what we are interested in determining
81
Concepts of Hypothesis Testing
  • There are two possible decisions that can be
    made
  • Conclude that there is enough evidence to support
    the alternative hypothesis
  • (also stated as rejecting the null hypothesis in
    favor of the alternative)
  • Conclude that there is not enough evidence to
    support the alternative hypothesis
  • (also stated as not rejecting the null
    hypothesis in favor of the alternative)
  • NOTE we do not say that we accept the null
    hypothesis

82
Concepts of Hypothesis Testing
  • Once the null and alternative hypotheses are
    stated, the next step is to randomly sample the
    population and calculate a test statistic (in
    this example, the sample mean).
  • If the test statistics value is inconsistent
    with the null hypothesis we reject the null
    hypothesis and infer that the alternative
    hypothesis is true.
  • For example, if were trying to decide whether
    the mean is not equal to 350, a large value of
    (say, 600) would provide enough evidence. If
    is close to 350 (say, 355) we could not say that
    this provides a great deal of evidence to infer
    that the population mean is different than 350.

83
Concepts of Hypothesis Testing
  • Two possible errors can be made in any test
  • A Type I error occurs when we reject a true null
    hypothesis and
  • A Type II error occurs when we dont reject a
    false null hypothesis.
  • There are probabilities associated with each type
    of error
  • P(Type I error)
  • P(Type II error )
  • is called the significance level.

84
Types of Errors
  • A Type I error occurs when we reject a true null
    hypothesis (i.e. Reject H0 when it is TRUE)
  • A Type II error occurs when we dont reject a
    false null hypothesis (i.e. Do NOT reject H0 when
    it is FALSE)

85
Types of Errors
  • Back to our example, we would commit a Type I
    error if
  • Reject H0 when it is TRUE
  • We reject H0 ( 350) in favor of H1 (
    ? 350) when in fact the real value of is
    350.
  • We would commit a Type II error in the case
    where
  • Do NOT reject H0 when it is FALSE
  • We believe H0 is correct ( 350), when in
    fact the real value of is something other
    than 350.

86
Recap
  • The null hypothesis must specify a single value
    of the parameter (e.g. ___)
  • Assume the null hypothesis is TRUE.
  • Sample from the population, and build a statistic
    related to the parameter hypothesized (e.g. the
    sample mean, )
  • Compare the statistic with the value specified in
    the first step

87
Example
  • A department store manager determines that a new
    billing system will be cost-effective only if the
    mean monthly account is more than 170.
  • A random sample of 400 monthly accounts is drawn,
    for which the sample mean is 178. The accounts
    are approximately normally distributed with a
    standard deviation of 65.
  • Can we conclude that the new system will be
    cost-effective?

88
Example
  • The system will be cost effective if the mean
    account balance for all customers is greater than
    170.
  • We express this belief as a our research
    hypothesis, that is
  • H1 gt 170 (this is what we want to
    determine)
  • Thus, our null hypothesis becomes
  • H0 170 (this specifies a single value
    for the parameter of interest)

89
Example
  • What we want to show
  • H1 gt 170
  • H0 170 (well assume this is true)
  • We know
  • n 400
  • 178
  • 65
  • Hmm. What to do next?!

90
Example
  • To test our hypotheses, we can use two different
    approaches
  • The rejection region approach (typically used
    when computing statistics manually), and
  • The p-value approach (which is generally used
    with a computer and statistical software).
  • We will explore both in turn

91
Example. Rejection Region
  • The rejection region is a range of values such
    that if the test statistic falls into that range,
    we decide to reject the null hypothesis in favor
    of the alternative hypothesis.

is the critical value of to reject H0.
92
Example
  • It seems reasonable to reject the null hypothesis
    in favor of the alternative if the value of the
    sample mean is large relative to 170, that is if
    gt .

P( gt ) is also P(rejecting H0
given that H0 is true) P(Type I error)
93
Example
  • All thats left to do is calculate and
    compare it to 178.

we can calculate this based on any level of
significance ( ) we want
94
Example
  • At a 5 significance level (i.e. 0.05), we
    get
  • Solving we compute 175.34
  • Since our sample mean (178) is greater than the
    critical value we calculated (175.34), we reject
    the null hypothesis in favor of H1, i.e. that
  • gt 170 and that it is cost effective
    to install the new billing system

95
Example The Big Picture
175.34
178
Reject H0 in favor of
96
Standardized Test Statistic
  • An easier method is to use the standardized test
    statistic
  • and compare its result to (rejection
    region z gt )
  • Since z 2.46 gt 1.645 (z.05), we reject H0 in
    favor of H1

97
p-Value
  • The p-value of a test is the probability of
    observing a test statistic at least as extreme as
    the one computed given that the null hypothesis
    is true.
  • In the case of our department store example, what
    is the probability of observing a sample mean at
    least as extreme as the one already observed
    (i.e. 178), given that the null hypothesis
    (H0 170) is true?

p-value
98
Interpreting the p-value
  • The smaller the p-value, the more statistical
    evidence exists to support the alternative
    hypothesis.
  • We observe a p-value of .0069, hence there is
    evidence to support H1 gt 170.

99
Interpreting the p-value
Overwhelming Evidence (Highly Significant)
Strong Evidence (Significant)
Weak Evidence (Not Significant)
No Evidence (Not Significant)
0 .01
.05 .10
p.0069
100
Interpreting the p-value
  • Compare the p-value with the selected value of
    the significance level
  • If the p-value is less than , we judge the
    p-value to be small enough to reject the null
    hypothesis.
  • If the p-value is greater than , we do not
    reject the null hypothesis.
  • Since p-value .0069 lt .05, we reject H0
    in favor of H1

101
Another example
  • The objective of the study is to draw a
    conclusion about the mean payment period. Thus,
    the parameter to be tested is the population
    mean. We want to know whether there is enough
    statistical evidence to show that the population
    mean is less than 22 days. Thus, the alternative
    hypothesis is
  • H1µ lt 22
  • The null hypothesis is
  • H0µ 22

102
Another example
  • The test statistic is
  • We wish to reject the null hypothesis in favor of
    the alternative only if the sample mean and hence
    the value of the test statistic is small enough.
    As a result we locate the rejection region in the
    left tail of the sampling distribution.
  • We set the significance level at 10.

103
Another example
  • Rejection region
  • Assume
  • and
  • p-value P(Z lt -.91) .5 - .3186 .1814

Conclusion There is not enough evidence to infer
that the mean is less than 22.
104
One and TwoTail Testing
  • The department store example was a one tail test,
    because the rejection region is located in only
    one tail of the sampling distribution
  • More correctly, this was an example of a right
    tail test.

105
One and TwoTail Testing
  • The payment period example is a left tail test
    because the rejection region was located in the
    left tail of the sampling distribution.

106
Right-Tail Testing
  • Calculate the critical value of the mean ( )
    and compare against the observed value of the
    sample mean ( )

107
Left-Tail Testing
  • Calculate the critical value of the mean ( )
    and compare against the observed value of the
    sample mean ( )

108
TwoTail Testing
  • Two tail testing is used when we want to test a
    research hypothesis that a parameter is not equal
    (?) to some value

109
Example
  • KPN argues that its rates are such that customers
    wont see a difference in their phone bills
    between them and their competitors. They
    calculate the mean and standard deviation for all
    their customers at 17.09 and 3.87
    (respectively).
  • They then sample 100 customers at random and
    recalculate a monthly phone bill based on
    competitors rates.
  • What we want to show is whether or not
  • H1 ? 17.09. We do this by assuming that
  • H0 17.09

110
Example
  • The rejection region is set up so we can reject
    the null hypothesis when the test statistic is
    large or when it is small.
  • That is, we set up a two-tail rejection region.
    The total area in the rejection region must sum
    to , so we divide this probability by 2.

stat is small
stat is large
111
Example
  • At a 5 significance level (i.e. .05), we
    have
  • /2 .025. Thus, z.025 1.96 and our
    rejection region is
  • z lt 1.96 -or- z gt 1.96

z
-z.025
z.025
0
112
Example
  • From the data, we calculate 17.55
  • Using our standardized test statistic
  • We find that
  • Since z 1.19 is not greater than 1.96, nor less
    than 1.96 we cannot reject the null hypothesis
    in favor of H1. That is there is insufficient
    evidence to infer that there is a difference
    between the bills of KPN and the competitor.

113
Summary of One- and Two-Tail Tests
114
Probability of a Type II Error
  • It is important that that we understand the
    relationship between Type I and Type II errors
    that is, how the probability of a Type II error
    is calculated and its interpretation.
  • Recall previous example
  • H0 170
  • H1 gt 170
  • At a significance level of 5 we rejected H0 in
    favor of H1 since our sample mean (178) was
    greater than the critical value of (175.34)

115
Probability of a Type II Error
  • A Type II error occurs when a false null
    hypothesis is not rejected.
  • In our example this means that if is less
    than 175.34 (our critical value) we will not
    reject our null hypothesis, which means that we
    will not install the new billing system.
  • Thus, we can see that
  • P( lt 175.34 given that the null
    hypothesis is false)

116
Example
  • P( lt 175.34 given that the null
    hypothesis is false)
  • The condition only tells us that the mean ? 170.
    We need to compute for some new value of
    . For example, suppose the mean account
    balance needs to be 180 in order to cost justify
    the new billing system
  • P( lt 175.34, given that 180),
    thus

117
Example
Our original hypothesis
our new assumption
118
Effects on of Changing
  • Decreasing the significance level ,
    increases the value of and vice versa.
  • Consider this diagram again. Shifting the
    critical value line to the right (to decrease
    ) will mean a larger area under the lower curve
    for (and vice versa)

119
Judging the Test
  • A statistical test of hypothesis is effectively
    defined by the significance level ( ) and
    the sample size (n), both of which are selected
    by the statistics practitioner.
  • Therefore, if the probability of a Type II error
    ( ) is judged to be too large, we can reduce
    it by
  • increasing , and/or
  • increasing the sample size, n.

120
Judging the Test
  • For example, suppose we increased n from a sample
    size of 400 account balances to 1,000
  • The probability of a Type II error ( ) goes
    to a negligible level while remains at 5

121
Judging the Test
  • The power of a test is defined as 1 .
  • It represents the probability of rejecting the
    null hypothesis when it is false.

122
Error Rates and Power(H0 and H1 null and
alternative hypothes)
123
Factors Affecting Power
  • Increasing overall sample size increases power
  • Having unequal group sizes usually reduces power
  • Larger size of effect being tested increases
    power
  • Setting lower significance level decreases power
  • Violations of assumptions underlying test often
    decrease power substantially

124
Exercises
  • Exercises see word document

125
The t-test
126
Recall t distribution.
  • Take random sample of size n from a N(m,s2)
    population.
  • has a standard normal
    distribution.
  • Consider .
  • This is approximately normal if n is large.
  • If n is small, S is not expected to be close to
    s. S introduces additional variability. Thus
    this statistic will be more variable that a
    standard normal random variable.
  • This statistic follows a t distribution with
    n-1degrees of freedom.

127
The t distribution.
red t with 1 d.f., green t with 5
d.f., yellow t with 10 d.f., blue standard
normal
The t distribution is similar in shape to the
normal distribution, but is more spread out. As
the degrees of freedom go to infinity the t
distribution approaches the standard normal
distribution.
128
Confidence Intervals.
  • Suppose that the population is normally
    distributed with mean m and variance s2. Then
  • If s is known, a 100(1-a) confidence interval
    for m is.
  • If s is not known, a 100(1-a) confidence
    interval for m is.

129
Overview of the t-test
  • The t-test is used to help make decisions about
    population values.
  • There are two main forms of the t-test, one for a
    single sample and one for two samples.
  • The one sample t-test is used to test whether a
    population has a specific mean value
  • The two sample t-test is used to test whether
    population means are equal, e.g., do training and
    control groups have the same mean.

130
One-sample t-test
  • We can use a confidence interval to test or
    decide whether a population mean has a given
    value.
  • For example, suppose we want to test whether the
    mean height of women at USF is less than 68
    inches.
  • We randomly sample 50 women students at USF.
  • We find that their mean height is 63.05 inches.
  • The SD of height in the sample is 5.75 inches.
  • Then we find the standard error of the mean by
    dividing SD by sqrt(N) 5.75/sqrt(50) .81.
  • The critical value of t with (50-1) df is
    2.01(find this in a t-table).
  • Our confidence interval is, therefore, 63.05
    plus/minus 1.63.

131
One-sample t-test example
Take a sample, set a confidence interval around
the sample mean. Does the interval contain the
hypothesized value?
132
One-sample t-test Example
The sample mean is roughly six standard
deviations (St. Errors) from the hypothesized
population mean. If the population mean is
really 68 inches, it is very, very unlikely that
we would find a sample with a mean as small as
63.05 inches.
133
Two-sample t-test
  • Used when we have two groups, e.g.,
  • Experimental vs. control group
  • Males vs. females
  • New training vs. old training method
  • Tests whether group population means are the
    same.
  • Can be means are just same or different
    (nondirectional)
  • or can predict one group higher (directional).

134
Sampling Distribution of Mean Differences
  • Suppose we sample 2 groups of size 50 at random
    from USF.
  • We measure the height of each person and find the
    mean for each group.
  • Then we subtract the mean for group 1 from the
    mean for group 2. Suppose we do this over and
    over.
  • We will then have a sampling distribution of mean
    differences.
  • If the two groups are sampled at random from 1
    population, the mean of the differences in the
    long run will be zero because the mean for both
    groups will be the same.
  • The standard deviation of the sampling
    distribution will be

The standard error of the difference is the root
of the sum of squared standard errors of the
mean.
135
Example of the Standard Error of the Difference
in Means
Suppose that at USF the mean height is 68 inches
and the standard deviation of height is 6 inches.
Suppose we sampled people 100 at a time into two
groups. We would expect that the average mean
difference would be zero. What would the
standard deviation of the distribution of
differences be?
The standard error for each group mean is .6, for
the difference in means, it is .85.
136
Estimating the Standard Error of Mean Differences
The USF scenario we just worked was based on
population information. That is
We generally dont have population values. We
usually estimate population values with sample
data, thus
All this says is that we replace the population
variance of error with the appropriate sample
estimators.
137
Pooled Standard Error
We can use this formula when the sample sizes for
the two groups are equal.
When the sample sizes are not equal across
groups, we find the pooled standard error. The
pooled standard error is a weighted average,
where the weights are the groups degrees of
freedom.
138
Back to the Two-Sample t
The formula for the two-sample t-test for
independent samples looks like this
This says we find the value of t by taking the
difference in the two sample means and dividing
by the standard error of the difference in means.

139
Example of the two-sample t, Empathy by College
Major
Suppose we have a professionally developed test
of empathy. The test has people view film
clips and guess what people in the clips are
feeling. Scores come from comparing what
people guess to what the people in the films said
they felt at the time. We want to know
whether Psychology majors have higher scores on
average to this test than do Physics majors.
No direction, we just want to know if there is
a difference. So we find some (N15) of each
major and give each the test.
140
Empathy Scores
141
Empathy
142
Exercise
  • Exercises t-test, see word document

143
Chi-square
144
  • Background
  • 1. Suppose there are n observations.
  • 2. Each observation falls into a cell (or class).
  • 3. Observed frequencies in each cell O1, O2, O3,
    , Ok.
  • Sum of the observed frequencies is n.
  • 4. Expected, or theoretical, frequencies E1, E2,
    E3, . . . , Ek.

145
  • Goal
  • 1. Compare the observed frequencies with the
    expected frequencies.
  • 2. Decide whether the observed frequencies seem
    to agree or seem to disagree with the expected
    frequencies.
  • Methodology
  • Use a chi-square statistic
  • Small values of c2 Observed frequencies close to
    expected frequencies.
  • Large values of c2 Observed frequencies do not
    agree with expected frequencies.

146
  • Sampling Distribution of c2
  • When n is large and all expected frequencies are
    greater than or equal to 5, then c2 has
    approximately a chi-square distribution.
  • Recall
  • Properties of the Chi-Square Distribution
  • 1. c2 is nonnegative in value it is zero or
    positively valued.
  • 2. c2 is not symmetrical it is skewed to the
    right.
  • 3. c2 is distributed so as to form a family of
    distributions, a separate distribution for each
    different number of degrees of freedom.

147
  • Critical values for chi-square
  • 1. See Table.
  • 2. Identified by degrees of freedom (df) and the
    area under the curve to the right of the critical
    value.
  • 3. c2(df, a) critical value of a chi-square
    distribution with df degrees of freedom and a
    area to the right.
  • 4. Chi-square distribution is not symmetrical
    critical values associated with right and left
    tails are given separately.

148
  • Example Find c2(16, 0.05).

Portion of Table
c2(16, 0.05) 26.3
149
  • Testing Procedure
  • 1. H0 The probabilities p1, p2, . . . , pk are
    correct.
  • Ha At least two probabilities are incorrect.
  • 2. Test statistic
  • 3. Use a one-tailed critical region the
    right-hand tail.
  • 4. Degrees of freedom df k - 1.
  • 5. Expected frequencies
  • 6. To ensure a good approximation to the
    chi-square distribution Each expected frequency
    should be at least 5

150
  • Example A market research firm conducted a
    consumer-preference experiment to determine which
    of 5 new breakfast cereals was the most appealing
    to adults. A sample of 100 consumers tried each
    cereal and indicated the cereal he or she
    preferred. The results are given in the
    following table
  • Is there any evidence to suggest the consumers
    had a preference for one cereal, or did they
    indicate each cereal was equally likely to be
    selected? Use a 0.05.

151
  • Solution
  • If no preference was shown, we expect the 100
    consumers to be equally distributed among the 5
    cereals. Thus, if no preference is given, we
    expect (100)(0.2) 20 consumers in each class.
  • 1. The Set-up
  • a. Population parameter of concern Preference
    for each cereal, the probability that a
    particular cereal is selected.
  • b. The null and alternative hypotheses
  • H0 There was no preference shown (equally
    distributed).
  • Ha There was a preference shown (not equally
    distributed).
  • 2. The Hypothesis Test Criteria
  • a. Assumptions The 100 consumers represent a
    random sample.
  • b. Test statistic c2 with df k - 1 5 - 1
    4
  • c. Level of significance a 0.05.

152
  • 3. The Sample Evidence
  • a. Sample information Table given in the
    statement of the problem.
  • b. Calculate the value of the test statistic
  • c2 3.2

153
  • 4. The Probability Distribution (Classical
    Approach)
  • a. Critical value c2(k - 1, 0.05) c2(4, 0.05)
    9.49
  • b. c2 is not in the critical region.
  • 4. The Probability Distribution (p-Value
    Approach)
  • a. The p-value
  • Using computer P 0.5429.
  • b. The p-value is larger than the level of
    significance, a.
  • 5. The Results
  • a. Decision Fail to reject H0.
  • b. Conclusion At the 0.05 level of
    significance, there is no evidence to suggest
    the consumers showed a preference for any one
    cereal.

154
  • r c Contingency Table
  • r number of rows c number of columns.
  • Used to test the independence of the row factor
    and the column factor.
  • Degrees of freedom
  • n grand total.
  • 5. Expected frequency in the ith row and the jth
    column
  • Each Ei,j should be at least 5.
  • 6. R1, R2, . . . , Rr and C1, C2, . . . Cc
    marginal totals.

155
  • Contingency table showing sample results and
    expected values

156
  • 4. The Probability Distribution (Classical
    Approach)
  • a. Critical value c2(4, 0.01) 13.3
  • b. c2 is in the critical region.
  • 4. The Probability Distribution (p-Value
    Approach)
  • a. The p-value
  • By computer P 0.0068.
  • b. The p-value is smaller than the level of
    significance, a.
  • 5. The Results
  • a. Decision Reject H0.
  • b. Conclusion There is evidence to suggest that
    opinion on tax reform and political party are
    not independent.

157
ANOVA
  • Analysis of Variance

158
From t to F
  • In the independent samples t test, you learned
    how to use the t distribution to test the
    hypothesis of no difference between two
    population means.
  • Suppose, however, that we wish to know about the
    relative effect of three or more different
    treatments?

159
From t to F
  • We could use the t test to make comparisons among
    each possible combination of two means.
  • However, this method is inadequate in several
    ways.
  • It is tedious to compare all possible
    combinations of groups.
  • Any statistic that is based on only part of the
    evidence (as is the case when any two groups are
    compared) is less stable than one based on all of
    the evidence.
  • There are so many comparisons that some will be
    significant by chance.

160
From t to F
  • What we need is some kind of survey test that
    will tell us whether there is any significant
    difference anywhere in an array of categories.
  • If it tells us no, there will be no point in
    searching further.
  • Such an overall test of significance is the F
    test, or the analysis of variance, or ANOVA.

161
The logic of ANOVA
  • Hypothesis testing in ANOVA is about whether the
    means of the samples differ more than you would
    expect if the null hypothesis were true.
  • This question about means is answered by
    analyzing variances.
  • Among other reasons, you focus on variances
    because when you want to know how several means
    differ, you are asking about the variances among
    those means.

162
Two Sources of Variability
  • In ANOVA, an estimate of variability between
    groups is compared with variability within
    groups.
  • Between-group variation is the variation among
    the means of the different treatment conditions
    due to chance (random sampling error) and
    treatment effects, if any exist.
  • Within-group variation is the variation due to
    chance (random sampling error) among individuals
    given the same treatment.

163
Variability Between Groups
  • There is a lot of variability from one mean to
    the next.
  • Large differences between means probably are not
    due to chance.
  • It is difficult to imagine that all six groups
    are random samples taken from the same
    population.
  • The null hypothesis is rejected, indicating a
    treatment effect in at least one of the groups.

164
Variability Within Groups
  • Same amount of variability between group means.
  • However, there is more variability within each
    group.
  • The larger the variability within each group, the
    less confident we can be that we are dealing with
    samples drawn from different populations.

165
The F Ratio
166
Two Sources of Variability
167
Two Sources of Variability
168
The F Ratio
mean squares between
mean squares within
169
The F Ratio
sum of squares between
sum of squares within
degrees of freedom within
degrees of freedom between
Sum of Squares
Degrees of Freedom
170
The F Ratio
sum of squares total
degrees of freedom total
171
The F Ratio SS Between
Find each group total, square it, and divide by
the number of subjects in the group.
Grand Total (add all of the scores together, then
square the total)
Total number of subjects.
172
The F Ratio SS Within
Square each individual score and then add up all
of the squared scores.
Squared group total.
Number of subjects in each group.
173
The F Ratio SS Total
Grand Total (add all of the scores together, then
square the total)
Square each score, then add all of the squared
scores together.
Total number of subjects.
174
An Example ANOVA
  • A study compared the intensity of pain among
    three groups of treatment.
  • Determine the significance of the difference
    among groups, using the .05 level of
    significance.
  • Treatment 1 Treatment 2 Treatment 3
  • 7 12 8
  • 6 8 10
  • 5 9 12
  • 6 11 10

175
An Example ANOVA
  • State the research hypothesis.
  • Do ratings of the intensity of pain differ for
    the three treatments?
  • State the statistical hypothesis.

176
Nondirectional Test
  • In testing the hypothesis of no difference
    between two means, a distinction was made between
    directional and nondirectional alternative
    hypotheses.
  • Such a distinction no longer makes sense when the
    number of means exceeds two.
  • A directional test is possible only in situations
    where there are only two ways (directions) that
    the null hypothesis could be false.
  • H0 may be false in any number of ways.
  • Two or more group means may be alike and the
    remainder differ, all may be different, and so on.

177
Degrees of Freedom
  • Between
  • Within

178
An Example ANOVA
  • Set decision rule.

179
An Example ANOVA
  • Set the decision rule.

180
An Example ANOVA
  • Calculate the test statistic.

Grand Total 104
181
An Example ANOVA
  • Calculate the test statistic.

Grand Total 104
182
An Example ANOVA
183
An Example ANOVA
  • Determine if your result is significant.
  • Reject H0, 9.61gt4.26
  • Interpret your results.
  • There is a significant difference between the
    treatments.
  • ANOVA Summary Table
  • In the literature, the ANOVA results are often
    summarized in a table.

Source df SS MS F Between Groups 2 42.67 21.34 9
.61 Within Groups 9 20 2.22 Total 11 62.67
184
After the F Test
  • When an F turns out to be significant, we know,
    with some degree of confidence, that there is a
    real difference somewhere among our means.
  • But if there are more than two groups, we dont
    know where that difference is.
  • Post hoc tests have been designed for doing
    pair-wise comparisons after a significant F is
    obtained.

185
Exercise 6 ANOVA
  • A psychologist interested in artistic preference
    randomly assigns a group of 15 subjects to one of
    three conditions in which they view a series of
    unfamiliar abstract paintings.
  • The 5 participants in the famous condition are
    led to believe that these are each famous
    paintings.
  • The 5 participants in the critically acclaimed
    condition are led to believe that these are
    paintings that are not famous but are highly
    thought of by a group of professional art
    critics.
  • The 5 in the control condition are given no
    special information about the paintings.
  • Does what people are told about paintings make a
    difference in how well they are liked? Use the
    .01 level of significance.

186
Linear and non-linear models
187
Review linear regression
  • Simplest form
  • Fit a straight line through
  • data points xi ,yi, i1....n, ngt2
  • y ax b
  • x predictor
  • y predicted value (outcome)
  • a slope
  • b y-axes intercept
  • Goal determine parameters a,b

188
Review linear regression
Find values for a and b such that sum of squared
error is minimized
189
Review linear regression
Predicted values yaxb Measurments y minimize
A minimum of a function (R) is characterized by a
zero first derivative with respect to the
parameters
190
Intermezzo minimum of function
191
Review linear regression
A minimum of a function (R) is characterized by a
zero first derivative with respect to the
parameters ? this provides the parameter values
for the model function
192
Review linear regression
a
Explicit expressions for parameters a and b!!
193
Linear and nonlinear models 1
  • (non) linear in the parameters (a, ß, ?)
  • Examples of linear models
  • yaßx (linear)
  • yaßx? x2 (polynomial)
  • yaßlog(x) (log)

194
Example
y varies linear with a for fixed x
195
Example
y varies linear with a for fixed x
196
Linear and nonlinear models 2
  • y ß0 ß1x1 ß2x2 e
  • -linear model (in parameters)
  • -y is linear combination of xs

-y is not a linear combination of xs -linear in
the parameters -We can use MLR if variables are
transformed x11/x1 x2x2 y ß0 ß1x1
ß2x2 e
197
Linear and nonlinear models 3
  • Models like
  • cannot be linearized and must be solved with
    nonlinear regression techniques

198
Linear and nonlinear models 4
  • Nonlinear model
  • At least one of the derivatives of the function
    wrt the parameters depends on at least one of the
    parameters (thus, slope of line at fixed x is not
    constant)

y ßlog(x) dy/dß log(x) Linear model
y ß0 ß1x1 ß2x2 dy/dß1 x1 Linear model
Nonlinear model
199
Significance testing and multiple testing
correction
200
Multiple testing
  • Say that you perform a statistical test with a
    0.05 threshold, but you repeat the test on twenty
    different observations.
  • Assume that all of the observations are
    explainable by the null hypothesis.
  • What is the chance that at least one of the
    observations will receive a p-value less than
    0.05?

201
Multiple testing
  • Say that you perform a statistical test with a
    0.05 threshold, but you repeat the test on twenty
    different observations. Assuming that all of the
    observations are explainable by the null
    hypothesis, what is the chance that at least one
    of the observations will receive a p-value less
    than 0.05?
  • Pr(making a mistake) 0.05
  • Pr(not making a mistake) 0.95
  • Pr(not making any mistake) 0.9520 0.358
  • Pr(making at least one mistake) 1 - 0.358
    0.642
  • There is a 64.2 chance of making at least one
    mistake.

202
Percentage sugar in candy (process 1)
Percentage sugar in candy (process 2)
no difference
statistical test (alpha0.05)
100 candy bars
100 candy bars
5 change of finding a difference (e.g. p0.003)
Suppose the company is required to do an
expensive tuning of process 2 if a difference is
found. They are willing to accept an Type 1 error
of 5. Thus only 5 of making wrong decision.
203
Percentage sugar in candy (process 1)
Percentage sugar in candy (process 2)
no difference
Day 1
statistical test (alpha0.05)
Day 2
statistical test (alpha0.05)
Change of 64.2 of finding at least
one significant difference Overall Type 1 error
64.2
Day 20
statistical test (alpha0.05)
204
Bonferroni correction
  • Assume that individual tests are independent.
  • Divide the desired p-value threshold by the
    number of tests performed.
  • For the previous example, 0.05 / 20 0.0025.
  • Pr(making a mistake) 0.0025
  • Pr(not making a mistake) 0.9975
  • Pr(not making any mistake) 0.997520 0.9512
  • Pr(making at least one mistake) 1 - 0.9512
    0.0488
  • meaning that the probability of one of the total
    number of tests being wrongfully said to be
    significantly different is of magnitude alpha
    (0.0488)
  • This is also known as correcting for the Family
    Wise Error (FWE). It is clear though that this
    highly increases the beta error (false negative),
    which is that many tests that should show an
    effect get below the corrected threshold.

205
Percentage sugar in candy (process 1)
Percentage sugar in candy (process 2)
no difference
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