Probability Theory

1
Probability Theory

• Review of essential concepts

2
Probability

• P(A ? B) P(A) P(B) P(A ? B)
• 0 P(A) 1
• P(O)1

3
Problem 1

• Given that P(A)0.6 and P(B)0.7, which of the
  following cannot be true?
• P(A ? B) 0.5 ? or
• P(A ? B) 0.9 ? and
• P(A ? B) 0.2
• P(A ? B) 0.4
• P(A ? B) 0.7

4
Conditional Probability

• A and B are called independent if P(A
  ? B) P(A) P(B)
• P(A B) P(A ? B)/P(B)
• P(A B) ???? A ? B
• A and B are independent ? P(AB)P(A)

5
Complete Probability

• P(A) P(AH1)P(H1)
• P(AH2)P(H2)
• P(AHn)P(Hn)
• H1, H2, Hn complete disjoint system of events

6
Bayes Formula

• P(BA) - prior probability
• P(AB) posterior probability

7
Problem 2

• Suppose a certain drug test is 99 sensitive and
  99 specific, that is, the test will correctly
  identify a drug user as testing positive 99 of
  the time, and will correctly identify a non-user
  as testing negative 98 of the time. Let's assume
  a corporation decides to test its employees for
  opium use, and 0.5 of the employees use the
  drug. What is the probability that, given a
  positive drug test, an employee is actually a
  drug user?

8
Problem 3

• We are presented with three doors - red, green,
  and blue - one of which has a prize. We choose
  the red door, which is not opened until the
  presenter performs an action. The presenter who
  knows what door the prize is behind, and who must
  open a door, but is not permitted to open the
  door we have picked or the door with the prize,
  opens the blue door and reveals that there is no
  prize behind it and subsequently asks if we wish
  to change our mind about our initial selection of
  red. What is the probability that the prize is
  behind each of the green and red doors?

9
Random Variables

• Discrete (Uniform, Binomial, Poisson, Geometric,
  Hypergeometric, Negative Binomial,)
• Continuous (Uniform, Normal, Exponential, Gamma,
  Chi-square, Student, Fisher, Dirchilet,)

10
Discrete Distributions
Poisson
11
Continuous Distributions
Beta distribution
12
Binomial Distribution

• Binomial random number the number of successes
  in n independent trials pprobability of success
  in one trial

p0.1
p0.3
p0.5
13
Problem 4
The probability that a certain machine will
produce a defective item is 0.20. If a
random sample of 6 items is taken from the output
of this machine, what is the probability
that there will be 5 or more defectives in the
sample?
14
Problem 5
There are 10 patients on the Neo-Natal Ward of a
local hospital who are monitored by 2 staff
members. If the probability (at any one time) of
a patient requiring emergency attention by a
staff member is 0.3, assuming the patients to be
behave independently, what is the probability at
any one time that there will not be sufficient
staff to attend all emergencies?
15
Cumulative Probability
X random variable F(x) P(X x)
Most of the data analysis tools have a built-in
function for the cumulative binomial probability
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Poisson Distribution

• Poisson random number the number of rare events
  per unit of time or space

?1.5
?5
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Problem 6

• The marketing manager of a company has noted that
  she usually receives 10 complaint calls during a
  week (consisting of five working days), and that
  the calls occur at random. Find the probability
  that she gets five such calls in one day.

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Problem 7

• The rate at which a particular defect occurs in
  lengths of plastic film being produced by a
  stable manufacturing process is 4.2 defects per
  75 meter length. A random sample of the film is
  selected and it was found that the length of the
  film in the sample was 25 meters. What is the
  probability that there will be at most 2 defects
  found in the sample?

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Normal Distribution
20
Cumulative Probability
Standard Normal Distribution
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Other Normal Distributions

• Z N(0,1)
• Mean 0
• Variance 1
• X N(µ, s)
• Mean µ
• Variance s2
• Z (X- µ)/s

22
Problem 8

• The diameters of steel disks produced in a plant
  are normally distributed with a mean of 2.5 cm
  and standard deviation of 0.02 cm. What is the
  probability that a disk picked at random has a
  diameter greater than 2.54 cm?

23
Problem 9

• The height of an adult male is known to be
  normally distributed with a mean of 69 inches and
  a standard deviation of 2.5 inches. What is the
  height of the doorway such that 96 percent of the
  adult males can pass through it without having to
  bend?

24
Problem 10

• The longevity of people living in a certain
  locality has a standard deviation of 14 years.
  What is the mean longevity if 30 of the people
  live longer than 75 years? Assume a normal
  distribution for life spans.

25
Normal Approximation to Binomial

• X Binom(n,p)
• n number of trials
• p probability of a single success
• X N(µ, s)
• µ np
• s2 np(1-p)

ngt40 npgt5 n(1-p)gt5
26
Problem 11
The unemployment rate in a certain city is 8.5 .
A random sample of 100 people from the labor
force is drawn. Find the approximate probability
that the sample contains at least ten unemployed
people.
27
Continuity correction
Normal approximation is still an approximation
28
Problem 12
Companies are interested in the demographics of
those who listen to the radio programs they
sponsor. A radio station has determined that only
20 of listeners phoning in to a morning talk
program are male. During a particular week, 200
calls are received by this program. What is the
approximate probability that at least 50 of the
callers are male?
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Poisson Approximation to Bionomial

• X Binom(n,p)
• n number of trials
• p probability of a single success
• X Poisson(?)
• ? np

n?8 p?0 np?const
30
Problem 13
A certain genetic characteristic will express
itself in 0.001 of the population. In a sample of
n3000 subjects, k7 are observed to display the
characteristic, whereas only three are expected
to display the characteristic. How likely is it
that a rate this great or greater could occur by
mere chance?
31
Expected Value
E(X) S xi pi not a random number
E(XY) 11/221/3 E(X)E(Y)
E(X) 01/211/21/2
E(Y) 01/312/32/3
X and Y are independent ? Xa and Yb are
independent events
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Variance
Var(X) E (X-E(X))2 E(X2)-(E (X))2
E(X)2/3
E(X-E(X)) -2/92/9 0
Var(X)4/91/31/92/32/9
E(X2)2/3
Var(X)E(X2)-E2(X)2/3 4/9 2/9
33
Expected Value and Variance

• X random variable
• E(XY) E(X) E(Y)
• E(cX) cE(X)
• E(c) c
• If X and Y are independent then E(XY) E(X)E(Y)
• Var(X)E(X2)-E2(X)
• Var(cX)c2Var(X)
• If X and Y are independent then Var(XY)
  Var(X)Var(Y)
• For arbitrary X and Y, Var(XY) Var(X) Var(Y)
  2Cov(X,Y)

34
Exercises

• Using properties of E(X) prove that
• Var(X) E (X-E(X))2 E(X2)-(E (X))2
• Var(XY) Var(X) Var(Y) 2Cov(X,Y)
• where
• Cov(X,Y)E (X-E(X))(Y-E(Y))
• Cov(X,Y)E(XY) - E(X)E(Y)
• Find X and Y such that X and Y are dependent but
  Cov(X,Y)0

35
Problem 14

• The Attila Barbell Company makes bars for weight
  lifting. The weights of the bars are independent
  and are normally distributed with a mean of 720
  ounces (45 pounds) and a standard deviation of 4
  ounces. The bars are shipped 10 in a box to the
  retailers. The weights of the empty boxes are
  normally distributed with a mean of 320 ounces
  and a standard deviation of 8 ounces. The weights
  of the boxes filled with 10 bars are expected to
  be normally distributed with a mean of 7,520
  ounces. What is the standard deviation?

36
Statistics

• Part I Sampling distribution

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Sampling Distribution

• Sample X1, X2, , Xn
• Xi are random numbers

Population heights of adult males

• All Xi are
• from the same distribution
• are independent

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Sample Mean

• All Xi are
• from the same distribution, i.e,
• E(Xi)µ, Var(Xi) s2
• are independent random numbers

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The Law of Large Numbers

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Illustrative example
Population 1,2,3, sample size n2
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Central Limit Theorem

• The sum of a sufficiently large number of
  identically distributed independent random
  variables is approximately normally distributed
  regardless of the population distribution

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Normal Approximation to Binomial
X number of successes in n trials XX1X2Xn
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Problem 18

• There are two games involving flipping a coin. In
  the first game you win a prize if you can throw
  between 45 and 55 of heads. In the second game
  you win if you can throw more than 80 heads. For
  each game would you rather flip the coin 30 times
  or 300 times?

44
Sampling distribution
X is approximately normal when ngt40 X is
approximately normal regardless of the
original distribution
45
Problem 15

• The average outstanding bill for delinquent
  customer accounts for a national department store
  chain is 187.50 with a standard deviation of
  54.50. In a simple random sample of 50
  delinquent accounts, what is the probability that
  the mean outstanding bill is over 200?

46
Problem 16

• The average number of daily emergency room
  admissions at a hospital is 85 with standard
  deviation of 37. In a simple random sample of 30
  days, what is the probability that the mean
  number of daily emergency admissions is between
  75 and 95?

47
Problem 17

• A summer resort rents rowboats to customers but
  does not allow more than four people to a boat.
  Each boat is designed to hold no more than 800
  pounds. Suppose the distribution of adult males
  who rent boats, including their clothes and gear,
  is normal with a mean of 190 pounds and standard
  deviation of 10 pounds. If the weights of
  individual passengers are independent, what is
  the probability that a group of four adult male
  passengers will exceed the acceptable weight
  limit of 800 pounds?

48
Statistics

• Part II Hypothesis testing

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Hypothesis testing

• H0 null hypothesis
• HA alternative hypothesis

In a court H0 the person is not guilty HA
the person is guilty Doctors appointment H0
patient is sick HA patient is not sick
50
Type I/II error

• Type I error (a)
• It is the error of rejecting a null hypothesis
  when it is actually true.
• Type II error (ß)
• It is the error of failing to reject a null
  hypothesis when it is in fact false.

51
Decision rule

• Assume we get many samples
• We set up a decision rule which rejects or
  accepts the hull hypothesis for each sample
• Sometimes we will commit Type I error
• Sometimes we will commit Type II error
• (Of course many times we will be correct!)

Decision rule comes separately from the set of
hypotheses
52
Type I/II error
53
Problem 19

• A patient claims that he consumes only 2000
  calories per day, but a dietician suspects that
  the actual figure is higher. The dietician plans
  to check his food intake for 30 days and will
  reject the patient's claim if the 30-day-mean is
  more than 2100 calories. If the standard
  deviation (in calories per day) is 350, what is
  the probability that the dietician will
  mistakenly reject a patient's true claim?

54
Problem 20

• City planners wish to test the claim that
  shoppers park for an average of only 47 minutes
  in the downtown area. The planners have decided
  to tabulate parking durations for 225 shoppers
  and to reject the claim if the sample mean
  exceeds 50 minutes. If the claim is wrong and the
  true mean is 51 minutes, what is the probability
  that the random sample will lead to a mistaken
  failure to reject the claim? Assume that the
  standard deviation in parking durations is 27
  minutes.

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P-value

• P-value is the probability of obtaining a result
  at least as extreme as the one that was actually
  observed, given that the null hypothesis is true.
• ???? ?? ??, ??? ?? ???????????? ? ???????
  ???????? ???? ?????, ?? ?????? ???? ??
  ??????????? ?????? ??, ??? ?? ????? ? ???????
  (???, ??? ??? ????)

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Hypothesis testing

• P-value is a function of sample
• a is a function of decision rule
• Reject H0 if p-valuelt a
• Small p-value indicates that you see something
  very unusual if H0 were true

57
Problem 21

• A service station advertises that its mechanics
  can change a muffler in only 15 minutes. A
  consumers group doubts this claim and runs a
  hypothesis test using 49 cars needing new
  mufflers. In this sample the mean changing time
  is 16.25 minutes with a standard deviation of 3.5
  minutes. Is this a strong evidence against the 15
  minute claim?

58
Estimators

• An estimator is a function of the observable
  sample data that is used to estimate an unknown
  population parameter
• is an estimator for µ
• s is an estimator for s
• is an estimator for p

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Unbiased effective estimators

• Let be the unknown parameter
• Let be an estimator
• is unbiased if
• is effective if

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Unbiased vs. effective
Unbiased but ineffective
Effective but biased
We are looking for unbiased and effective
estimators
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Mean Squared Error

• Bias
• Variance
• Mean Squared Error

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Problem ?

• A box contains 70 black and 30 white balls. Ten
  balls are chosen at random and two estimators of
  the following form are considered
• where n10. Which estimator is more effective?
  (i.e., has a smaller MSE?)

63
Standard error

• Standard error standard deviation of the
  estimator

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Problem 22

• A local restaurant owner claims that only 15 of
  visiting tourists stay for more than 2 days. A
  chamber of commerce volunteer is sure that the
  real percentage is higher. He plans to survey 100
  tourists and intends to speak up if at least 18
  of the tourists stay longer than 2 days. What is
  the probability of mistakenly rejecting the
  restaurant owner's claim if it is true?

65
Two-sample mean

• Two independent samples, X1,, Xn and Y1,,Ym
  have independent sample means

66
Two-sample proportion

• Two independent sample proportions

67
Problem 23

• A historian believes that the average height of
  soldiers in World War II was greater than that of
  soldiers in World War I. She examines a random
  sample of records of 100 men in each war and
  notes standard deviations of 2.5 and 2.3 inches
  in World War I and World War II, respectively. If
  the average height from the sample of World War
  II soldiers is 1 inch greater than that from the
  sample of World War I soldiers, what conclusion
  is justified from a two-sample hypothesis test
  where H0 µ1 µ2 vs. HA µ1lt µ2?

68
Confidence intervals

• Hypothesis testing A coffee machine is supposed
  to deliver 8 ounces of coffee in a cup, but in my
  sample of 10 cups I get only 7.5 ounces. Is this
  ok?
• Confidence intervals My sample of 10 cups of
  coffee contains on average 7.5 ounces of liquid.
  What is the likely estimate for the mean amount
  of coffee per cup?
• Hypothesis testing and construction of confidence
  intervals are mutually inverse problems

69
Confidence intervals

• Parameter Estimate critical SE,
• SE standard error

70
Critical value
0.025
0.025
0.95
z1.96
71
Problem 19 revisited

• A patient claims that he consumes only 2000
  calories per day, but a dietician suspects that
  the actual figure is higher. The dietician
  checked his food intake for 30 days and found
  that the 30-day-mean is more than 2100 calories.
  What is the 95 confidence interval for the
  number of calories in patients diet?
• Assume standard deviation of 350 calories per
  day.

72
Problem 22 revisited

• A chamber of commerce volunteer is interested in
  the percentage of visiting tourists staying for
  more than 2 days in a certain hotel. He surveyed
  100 tourists and found that 18 of them stay
  longer than 2 days. What is the 99 confidence
  interval for the percentage of visiting tourists
  who stay for more than 2 days?

73
Problem 24

• In a random sample of 300 high school students,
  225 said they managed time effectively, while in
  a similar sample of 270 college students, only
  108 felt they were effective time managers. What
  is a 99 confidence interval estimate for the
  difference between the proportions of high school
  and colleges students who think they manage time
  effectively?

74
Problem 25

• A medical researcher believes that taking 1000
  milligrams of vitamin C per day will result in
  fewer colds than a daily intake of 500 milligrams
  will. In a group of 50 volunteers taking 1000
  milligrams per day, the numbers of colds per
  individual during a winter season averaged 1.8
  with a variance of 1.5. Similar data from a group
  of 60 volunteers taking 500 milligrams per day
  showed an average of 2.4 with a variance of 1.6.
  What was the P-value of this test?

75
How do we get s?

• Population standard deviation is usually unknown
• If sample size is large (ngt40) then we can assume
  that the sample standard deviation (s)
  approximates the population standard deviation
  (s) well enough
• If sample size is small then this assumption is
  no longer valid, i.e., sampling error in the
  estimation of s cannot be ignored

76
Known vs. unknown s
s
known
unknown
z
Small sample
Large sample
t
z
77
Student t-distribution
78
Student t-distribution

• Student t-distribution has one parameter called
  degrees of freedom

• When the number of degrees of freedom is large,
  the t-distribution is close to z-distribution

79
t-distribution table
Degrees of freedom sample size - 1
80
Problem 26

• An article ("Undergraduate Marijuana use and
  Anger" by Sue Stoner) in a 1988 issue of the
  Journal of Psychology (Vol. 122, p. 33) reported
  that in a sample of 17 marijuana users the mean
  and standard deviation on an anger expression
  scale were 42.72 and 6.05, respectively. Test
  whether this result is significantly greater than
  the established mean of 41.6 for nonusers. What
  assumptions are necessary for the above test to
  be valid?

81
T-test assumptions

• Random sampling (like in z-test)
• Normal population (unlike z-test, where sample
  mean is automatically normal regardless of the
  population when sample size is large)
• Degrees of freedom number of independent
  observations (actually, residuals)

82
Problem 27

• A hospital exercise laboratory technician notes
  the resting pulse rates of five joggers to be 60,
  58, 59, 61, and 67, respectively, while the
  resting pulse rates of seven non-exercisers are
  83, 60, 75, 71, 91, 82, and 84, respectively.
  Establish a 99 confidence interval estimate for
  the difference in pulse rates between joggers and
  non-exercisers.
• (Means and standard deviations are 61, 78, 3.54,
  and 10.23, respectively)

83
Equal variances assumption

• Assume that both populations have the same
  standard deviation (i.e., amount of exercise
  affects mean of the population, not its standard
  deviation)

d.f. minn,m-1
d.f. n m - 2
84
Problem 27 revisited

• A hospital exercise laboratory technician notes
  the resting pulse rates of five joggers to be 60,
  58, 59, 61, and 67, respectively, while the
  resting pulse rates of seven non-exercisers are
  83, 60, 75, 71, 91, 82, and 84, respectively.
  Establish a 99 confidence interval estimate for
  the difference in pulse rates between joggers and
  non-exercisers. Assume equal variances.
• (Means and standard deviations are 61, 78, 3.54,
  and 10.23, respectively)

85
Problem 28

• A researcher believes a new diet should improve
  weight gain in laboratory mice. If ten control
  mice on the old diet gain an average of 4 ounces
  with a standard deviation of 0.3 ounces, while
  the average gain for the ten mice on the new diet
  is 4.8 ounces with a standard deviation of 0.2
  ounces, what is the p-value?

86
Dependent samples

• Trace metals in drinking water wells affect the
  flavor of the water and unusually high
  concentrations can pose a health hazard. In the
  paper, Trace Metals of South Indian River
  Region (Environmental Studies, 1982, 62-6),
  trace metal concentrations (mg/L) on zinc were
  found from water drawn from the bottom and the
  top of each of 6 wells.

87
Dependent samples
One sample t-test
88
FAQs

• Do I have to divide by square root of n?
• Yes, if you are looking for P(Xgt100)
• No, if you are looking for P(Xgt100)
• Do I have to divide by square root of n in
  one-proportion or two-proportion tests?
• No. If you use Standard Error, it already
  contains the square root of n
• When I compute standard deviation from the
  sample, do I have to divide it by square root of
  n?
• Yes, if your calculations involve sample mean.

89
Common misconception

• Sample standard deviation is an estimator for the
  population standard deviation
• Standard deviation of the sampling distribution
  is smaller than the population standard deviation
• Sample standard deviation is NOT an estimator for
  the standard deviation of the sampling
  distribution

90
Estimation of s
91
Chi-square table
92
Problem 29

• A supplier of 100 ohm/cm silicon wafers claims
  that his fabrication process can produce wafers
  with sufficient consistency so that the standard
  deviation of resistance for the lot does not
  exceed 10 ohm/cm. A sample of 10 wafers taken
  from the lot has a standard deviation of 13.97
  ohm/cm. Is the suppliers claim reasonable?

93
Problem 30

• A container of oil is supposed to contain 1000 ml
  of oil. We want to be sure that the standard
  deviation of the oil container is less than 20
  ml. We randomly select 10 cans of oil with a mean
  of 997 ml and a standard deviation of 32 ml.
  Using these sample construct a 95 confidence
  interval for the true value of sigma. Does the
  confidence interval suggest that the variation in
  oil containers is at an acceptable level?

94
Estimation of sample size

• What is a minimum sample size needed to estimate
  the population mean within 2 units?
• What is a minimum sample size needed to estimate
  the population proportion within 2 percent units?

95
Problem 31

• An electrical firm which manufactures a certain
  type of bulb wants to estimate its mean life.
  Assuming that the life of the light bulb is
  normally distributed and that the standard
  deviation is known to be 40 hours, how many bulbs
  should be tested so that we can be 90 percent
  confident that the estimate of the mean will not
  differ from the true mean life by more than 10
  hours?

96
Problem 32

• A quality control engineer wants to estimate the
  fraction of defective bulbs in a large lot of
  light bulbs. From past experience, he feels that
  the actual fraction of defective bulbs should be
  somewhere around 0.2 . How large a sample should
  be taken if he wants to estimate the true
  fraction within .02 using a 95 confidence
  interval?

97
Problem 33

• Many television viewers express doubts about the
  validity of certain commercials. Let p represent
  the true proportion of consumers who believe what
  is shown in Timex television commercials. If
  Timex has no prior information regarding the true
  value of p, how many consumers should be included
  in their sample so that they will be 85
  confident that their estimate is within 0.03 of
  the true value of p?

98
Statistics

• Part III Contingency tables

99
Non-parametric hypotheses

• H0 features are independent
• HA features are dependent

A restaurant owner surveys a random sample of 385
customers to determine whether customer
satisfaction is related to gender and age.
100
Assumption of independence
If gender/age and satisfaction were independent
then P(satisfied and young male)
P(satisfied)P(young male) P(satisfied)
302/385 P(young male) 33/385 P(satisfied and
young male) 30233/3852 Expected number of
satisfied young males 30233/385
101
Observed and Expected
Observed
Expected
102
Chi-square test for independence
d.f. (n-1)x(m-1)
103
Problem 34

• A sociologist conducts a test whether there is a
  relationship between cheating on exams and
  socioeconomic status. A random sample of 750 high
  school students yields the following results
• What is the conclusion about cheating and
  socioeconomic status at the 5 significance level?

104
Chi-square goodness of fit

• A grocery store manager wishes to determine
  whether a certain product will sell equally well
  in any of the five locations in the store. Five
  displays are set up, one for each location, and
  the resulting numbers of the product sold are
  noted
• Is there enough evidence to claim a difference?

105
Chi-square goodness of fit
Total 432948206 We expect 206/541.2 units
sold in each location H0 The distribution is
uniform HA The distribution is not uniform
d.f. n-1
106
Problem 35

• A geneticist claims that four species of fruit
  flies should appear in the ratio of 1339.
  Suppose that a sample of 4000 fruit flies
  contained 226, 764, 733, and 2277 flies of each
  species, respectively. At the 10 significance
  level, is there sufficient evidence to reject the
  geneticists hypothesis?

107
Chi-square test warning

• Chi-square test is applicable only if the
  expected value in each cell is greater than 5
  (Compare to Binomial Distribution)
• If this doesnt hold, you might find Fisher exact
  test more useful

108
Problem 36

• A sample of teenagers might be divided into male
  and female on the one hand, and those that are
  and are not currently dieting on the other. We
  hypothesize, perhaps, that the proportion of
  dieting individuals is higher among the women
  than among the men, and we want to test whether
  any difference of proportions that we observe is
  significant.

Expected lt 5
109
Fisher exact test
Hypergeometric Distribution
110
Statistics

• Part IV Regression and ANOVA

111
The least squares line

• A simple data set consists of data pairs
  (xi, yi), i 1, ..., n,
• where xi is an independent variable and yi is a
  dependent variable
• The model function has the form
• y a bx
• We wish to find a and b for which the model
  "best" fits the data.

112
Residuals

• The least squares method defines "best" as when S
  S ri2 is at minimum.
• A residual ri is defined as the difference
  between the values of the dependent variable and
  the predicted values from the estimated model
• ri yi - (a b xi)

113
Regression Line

• Residuals are shown by blue lines
• Sum of squares of the residuals is at minimum

114
Residual plot

• The sum of the residuals is always zero
• A pattern in the residual plot indicates that a
  non-linear model should be used

115
Influential scores and outliers

• In regression, an outlier is a data point with
  large residual
• An influential score is the data point which
  significantly influences the regression line
• If an influential score is removed from the
  sample, the regression line will change
  significantly

116
Problem 37

• Which of the five points is an outlier, and which
  is an influential score?

117
Solving the regression
118
Regression slope and intercept
119
Correlation Coefficient

• The correlation coefficient indicates the degree
  of linear dependence

120
Correlation and slope
121
Coefficient of determination

• SST total sum of squares
• SSX sum of squares explained by X
• SSE sum of squares of residuals
• SST SSXSSE
• The square of the sample correlation coefficient,
  which is also known as the coefficient of
  determination, is the fraction of the variance in
  y that is accounted for by a linear fit of x

122
Sums of squares
red
blue
123
SE of the regression slope

• The regression line is a result of random
  sampling
• Different samples produce different lines
• There is a family of lines for the given
  population you get just one

124
SE of the regression slope
where se is the standard deviation of the
regression error
125
Problem 38

• What is the equation of the fitted line?
• Find an approximate confidence interval for the
  regression slope?
• Test the hypothesis that the slope is non-zero

126
Problem 39
Find the regression line and a 95 confidence
interval for the regression slope.
127
Confidence vs. prediction intervals

• Suppose I fuel my car 7 days a week, from Sunday
  to Sunday, each day at a randomly chosen gas
  station. I get a sample of gasoline prices for 7
  days
• Confidence interval is for the average gasoline
  price on Monday
• Prediction interval is for a gasoline price at a
  randomly chosen gas station on Monday

128
Confidence vs. prediction intervals

• Confidence interval
• Prediction interval

129
Problem 39 revisited
Find the a 95 prediction interval for the dive
duration at 25 degrees Celsius
130
ANOVA Analysis of Variance

• A collection of models, in which the variance of
  the observed set is partitioned into components
  due to explanatory variables
• Assumptions
• Independence of observations
• The distributions in each of the groups are
  normal
• Variance homogeneity, called homoscedasticity
  the variance of data in groups should be the
  same.

131
One-way ANOVA

• A manager wishes to determine whether the mean
  times required to complete a certain task differ
  for the three levels of employee training. He
  randomly selected 10 employees with each of the
  three levels of training.
• Do the data provide sufficient evidence to
  indicate that the mean times required to complete
  a certain task differ for at least two of the
  three levels of training?

132
Steiners Theorem
a
xi
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133
Problem 40

• Three different milling machines were being
  considered for purchase by a manufacturer.
  Potentially, the company would be purchasing
  hundreds of these machines, so it wanted to make
  sure it made the best decision. Initially, five
  of each machine were borrowed, and each was
  randomly assigned to one of 15 technicians (all
  technicians were similar in skill). Each machine
  was put through a series of tasks and rated using
  a standardized test. The higher the score on the
  test, the better the performance of the machine.
  The data are

134
Partition of sum of squares

• SST SSA SSE
• SST total sum of squares
• SSA sum of squares for factor A
• SSE sum of squares of errors

135
Partition of sum of squares
136
The ANOVA table

• SSA Sum of squares Factor
• SSE Sum of squares Error
• MSA Mean sum of squares Factor
• MSE Mean sum of squares Error

137
Fisher distribution
138
Problem 40 solution
In EXCEL Tools -gt Data Analysis -gt Single Factor
ANOVA
139
Two-way ANOVA

• Group A is given vodka, Group B is given gin, and
  Group C is given a placebo. Groups are tested
  with a memory task. One-way ANOVA
• In an experiment testing the effects of
  expectations, subjects are randomly assigned to
  four groups
• expect vodkareceive vodka
• expect vodkareceive placebo
• expect placeboreceive vodka
• expect placeboreceive placebo
• Each group is then tested on a memory task.
  Two-way ANOVA

140
Partition of sum of squares

• SST SSA SSB SSE
• SST total sum of squares
• SSA sum of squares for factor A
• SSB sum of squares for factor B
• SSE sum of squares of errors

141
Partition of sum of squares
142
Problem 41

• Three different milling machines were being
  considered for purchase by a manufacturer.
  Machines are operated by 5 different crew
  technicians

143
What is the error term?
144
Two-way ANOVA table

• At 5 level
• The variation across rows (crew technicians) is
  NOT significant
• The variation across columns (machines) is
  significant

145
Problem 42

• Some varieties of nematodes feed on the roots of
  lawn grasses and crops such as strawberries and
  tomatoes. Four brands of nematocides are to be
  compared. Twelve plots of land of comparable
  fertility that were suffering from nematodes were
  planted with a crop. The yields of each plot were
  recorded and part of the ANOVA table appears
  below
• Find the value of F

146
THE END
147
Extra Problems

• All bags entering a research facility are
  screened. Ninety-seven percent of the bags that
  contain forbidden material trigger an alarm.
  Fifteen percent of the bags that do not contain
  forbidden material also trigger the alarm. If 1
  out of every 1,000 bags entering the building
  contains forbidden material, what is the
  probability that a bag that triggers the alarm
  will actually contain forbidden material?

148
Extra problems

• Pepper plants watered lightly every day for a
  month show an average growth of 27 cm with the
  standard deviation of 8.3 cm, while pepper plants
  watered heavily once a week for a month show an
  average growth of 29 cm with the standard
  deviation of 7.9 cm. In a sample of 60 plants,
  half of which were given each of the water
  treatments, what is the probability that the
  difference in average growth between the two
  halves is between -3 and 3 cm?