Confidence Intervals

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Confidence Intervals
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(1 a)100 Confidence Intervals
1. Mean, m, of a Normal population (s known).
2. Mean, m, of a Normal population (s unknown).
3. Variance, s2, of a Normal population.
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4. Standard deviation, s, of a Normal
population.
5. Success-failure (Bernoulli) probability, p.
6. General parameter, q.
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Hypothesis Testing

• An important area of statistical inference

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Definition

• Hypothesis (H)
• Statement about the parameters of the population
• In hypothesis testing there are two hypotheses of
  interest.
• The null hypothesis (H0)
• The alternative hypothesis (HA)

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• Either
• null hypothesis (H0) is true or
• the alternative hypothesis (HA) is true.
• But not both
• We say that are mutually exclusive and
  exhaustive.

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• One has to make a decision
• to either to accept null hypothesis (equivalent
  to rejecting HA)
• or
• to reject null hypothesis (equivalent to
  accepting HA)

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• There are two possible errors that can be made.
• Rejecting the null hypothesis when it is true.
  (type I error)
• accepting the null hypothesis when it is false
  (type II error)

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• An analogy a jury trial
• The two possible decisions are
• Declare the accused innocent.
• Declare the accused guilty.

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• The null hypothesis (H0) the accused is
  innocent
• The alternative hypothesis (HA) the accused is
  guilty

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• The two possible errors that can be made
• Declaring an innocent person guilty.
• (type I error)
• Declaring a guilty person innocent.
• (type II error)
• Note in this case one type of error may be
  considered more serious

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Decision Table showing types of Error
H0 is True
H0 is False


Correct Decision
Type II Error
Accept H0
Correct Decision
Type I Error
Reject H0
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• To define a statistical Test we
• Choose a statistic (called the test statistic)
• Divide the range of possible values for the test
  statistic into two parts
• The Acceptance Region
• The Critical Region

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• To perform a statistical Test we
• Collect the data.
• Compute the value of the test statistic.
• Make the Decision
• If the value of the test statistic is in the
  Acceptance Region we decide to accept H0 .
• If the value of the test statistic is in the
  Critical Region we decide to reject H0 .

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• Example
• We are interested in determining if a coin is
  fair.
• i.e. H0 p probability of tossing a head ½.
• To test this we will toss the coin n 10 times.
• The test statistic is x the number of heads.
• This statistic will have a binomial distribution
  with p ½ and n 10 if the null hypothesis is
  true.

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Sampling distribution of x when H0 is true
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• Note
• We would expect the test statistic x to be around
  5 if H0 p ½ is true.
• Acceptance Region 3, 4, 5, 6, 7.
• Critical Region 0, 1, 2, 8, 9, 10.
• The reason for the choice of the Acceptance
  region
• Contains the values that we would expect for x if
  the null hypothesis is true.

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• Definitions For any statistical testing
  procedure define
• a PRejecting the null hypothesis when it is
  true P type I error
• b Paccepting the null hypothesis when it is
  false P type II error

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• In the last example
• a P type I error p(0) p(1) p(2) p(8)
  p(9) p(10) 0.109, where p(x) are binomial
  probabilities with p ½ and n 10 .
• b P type II error p(3) p(4) p(5) p(6)
  p(7), where p(x) are binomial probabilities
  with p (not equal to ½) and n 10. Note these
  will depend on the value of p.

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Table Probability of a Type II error, b vs. p
Note the magnitude of b increases as p gets
closer to ½.
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The Power function of a Statistical Test

• Definition The Power Function, P(q1, ,qq) of a
  statistical test is defined as follows

P(q1, ,qq) P test rejects H0
Note if H0 is true P(q1, ,qq) Prejects H0
PType I error a.
if H0 is false P(q1, ,qq) Prejects H0
1- PType II error 1 - b.
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Graph of the Power function P(q1, ,qq)
b
Power function of ideal test
a
H0 is false
q
H0 is false
H0 is true
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Example n 10, Critical Region 0,1,2,8,9,10
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• Comments
• You can control a P type I error and b P
  type II error by widening or narrowing the
  acceptance region. .
• Widening the acceptance region decreases a P
  type I error but increases b P type II
  error.
• Narrowing the acceptance region increases a P
  type I error but decreases b P type II
  error.

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• Example Widening the Acceptance Region
• Suppose the Acceptance Region includes in
  addition to its previous values 2 and 8 then a
  P type I error p(0) p(1) p(9) p(10)
  0.021, where again p(x) are binomial
  probabilities with p ½ and n 10 .
• b P type II error p(2) p(3) p(4) p(5)
  p(6) p(7) p(8). Tabled values of are given
  on the next page.

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Table Probability of a Type II error, b vs. p
Note Compare these values with the previous
definition of the Acceptance Region. They have
increased,
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Example n 10, Critical Region 0,1, 9,10
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• Example Narrowing the Acceptance Region
• Suppose the original Acceptance Region excludes
  the values 3 and 7. That is the Acceptance Region
  is 4,5,6. Then a P type I error p(0)
  p(1) p(2) p(3) p(7) p(8) p(9) p(10)
  0.344.
• b P type II error p(4) p(5) p(6) .
  Tabled values of are given on the next page.

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Table Probability of a Type II error, b vs. p
Note Compare these values with the otiginal
definition of the Acceptance Region. They have
decreased,
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Example n 10, Critical Region
0,1,2,3,7,8,9,10
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Acceptance Region 4,5,6.
Acceptance Region 2,3,4,5,6,7,8.
Acceptance Region 3,4,5,6,7.
a 0.344
a 0.109
a 0.021
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Example n 10
A - Critical Region 0,1,2,8,9,10
B - Critical Region 0,1, 9,10
C - Critical Region 0,1,2,3,7,8,9,10
C
A
B
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• The Approach in Statistical Testing is
• Set up the Acceptance Region so that a is close
  to some predetermine value (the usual values are
  0.05 or 0.01)
• The predetermine value of a (0.05 or 0.01) is
  called the significance level of the test.
• The significance level (size) of the test is a
  Ptest makes a type I error

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The z-test for the Mean of a Normal Population

• We want to test, m, denote the mean of a normal
  population

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• The t distribution
• Estimating m, the mean of a Normal population
• (s2 unknown)

Let x1, , xn denote a sample from the normal
distribution with mean m and variance s2. Both m
and s2 are unknown
Recall
Also
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Situation
Let x1, , xn denote a sample from the normal
distribution with mean m and variance s2. Both m
is unknown and s2 is known

• We want to test
• H0 m m 0 (some specified value of m)
• Against
• HA

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The Data

• Let x1, x2, x3 , , xn denote a sample from a
  normal population with mean m and standard
  deviation s.
• Let
• we want to test if the mean, m, is equal to some
  given value m0.
• Obviously if the sample mean is close to m0 the
  Null Hypothesis should be accepted otherwise the
  null Hypothesis should be rejected.

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The Test Statistic

• To decide to accept or reject the Null Hypothesis
  (H0) we will use the test statistic

• If H0 is true we should expect the test statistic
  z to be close to zero.

• If H0 is true we should expect the test statistic
  z to have a standard normal distribution.

• If HA is true we should expect the test statistic
  z to be different from zero.

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• The sampling distribution of z when H0 is true
• The Standard Normal distribution

Accept H0
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• The Acceptance region

Accept H0
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• Acceptance Region
• Accept H0 if

• Critical Region
• Reject H0 if

• With this Choice

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Summary

• To Test for a binomial probability m
• H0 m m0 (some specified value of m)
• Against
• HA

• Decide on a PType I Error the significance
  level of the test (usual choices 0.05 or 0.01)

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1. Collect the data

• Compute the test statistic

• Make the Decision

• Accept H0 if

• Reject H0 if

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Example

• A manufacturer Glucosamine capsules claims that
  each capsule contains on the average

• 500 mg of glucosamine

To test this claim n 40 capsules were selected
and amount of glucosamine (X) measured in each
capsule.
Summary statistics
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• We want to test

Manufacturers claim is correct
against
Manufacturers claim is not correct
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The Test Statistic
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The Critical Region and Acceptance Region
Using a 0.05
za/2 z0.025 1.960
We accept H0 if -1.960 z 1.960
reject H0 if z lt -1.960 or z gt 1.960
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The Decision
Since z -2.75 lt -1.960 We reject H0
Conclude the manufacturer's claim is incorrect