PRED 354 TEACH. PROBILITY

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PRED 354 TEACH. PROBILITY STATIS. FOR PRIMARY
MATH

• Lesson 7
• Continuous Distributions 

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Hints

• Suppose that a school band.

One class is not included
Three classes are not included Or consisting of
only one class
Two classes are not included
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Hints
These are disjoint
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Corrections

• Find the probability of the subset of points such
  that

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Question

• Two boys A and B throw a ball at a target.
  Suppose that the probability that boy A will hit
  the target on any throw is 1/3 and the
  probability that boy B will hit the target on any
  throw is ¼. Suppose also that boy A throws first
  and the two boys take turns throwing. Determine
  the probability that the target will be hit for
  the first time on the third throw of boy A.

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Question

• If A and B are independent events and Pr(B)lt1,

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Question

• Suppose that a random variable X has discrete
  distribution with the following probability
  function
• Find the value of the constant

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The probability density function (p.d.f.)

• Every p.d.f f must satisfy the following two
  requirements
• Ex Suppose that X has a binomial with n2 and
  p1/2. Find f(x) and

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Example

• EX Suppose that the p.d.f of a certain random
  variable X is as follows
• Find the value of a constant c and sketch the
  p.d.f.
• Find the value of
• Sketch probability distribution function

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Normal p.d.f.
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Example

• EXLet we have a normal distribution with mean 0
  and variance 1.
• Find

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Example

• Adult heights form a normal distribution with a
  mean of 68 inches and standard deviation of 6
  inches.
• Find the probability of randomly selecting
  individual from this population who is taller
  than 80 inches?

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The distribution of sample means

• The distribution of sample means is the
  collection of sample means for all the possible
  random samples of a particular size (n) that can
  be obtained from a population.
• A sampling distribution is a distribution of
  statistics obtained by selecting all the possible
  samples of a specific size from a population.

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The distribution of sample means

• EX population 1, 3, 5, 7
• Sample size 2,

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The standard error of

• The standard deviation of the distribution of
  sample means is called the standard error of
• The standard deviation of the population
• The sample size

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Example

• A population of scores is normal, with µ50 and
  s12. Describe the distribution of sample means
  for samples size n16 selected from this
  population
• Shape?
• Mean?

• The distribution of samples will be almost
  perfectly normal if either one of the following
  two conditions is satisfied
• The population from which the samples are
  selected is normal distribution.
• The number scores (n) in each sample is
  relatively large, around 30 or more.

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Example

• EX A skewed distribution has µ60 and s8.
• What is the probability of obtaining a sample
  mean greater than 62 for a sample of n4?
• What is the probability of obtaining a sample
  mean greater than 62 for a sample of n64?

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• Introduction to hypothesis testing

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Hypothesis testing

• HP is an inferential procedure that uses sample
  data to evaluate the credibility of a hypothesis
  about a population.
• Using sample data as the basis for making
  conclusions about population
• GOAL to limit or control the probability of
  errors.

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Hypothesis testing (Steps)

• State the hypothesis
• H0 predicts that the IV has no effect on
  the DV for the population
• H0 Using constructivist method has no effect on
  the first graders math achievement.
• H1predicts that IV will have an effect on the
  DV for the population

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Hypothesis testing

• Setting the criteria for a decision
• The researcher must determine whether the
  difference between the sample data and the
  population is the result of the treatment effect
  or is simply due to sampling error.
• He or she must establish criteria (or cutoffs)
  that define precisely how much difference must
  exist between the data and the population to
  justify a decision that H0 is false.

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Hypothesis testing

• Collecting sample data
• Evaluating the null hypothesis
• The researcher compares the data with the
  null hypothesis (µ) and makes a decision
  according to the criteria and cutoffs that were
  established before.
• Decision
• reject the null hypothesis
• fail to reject the null hypothesis

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Errors in hypothesis testing
ACTUAL SITUATION ACTUAL SITUATION
No effect, H0 True Effect Exists, H0 False
Researcher decision Reject H0 Type I error Decision correct
Researcher decision Retain H0 Decision correct Type II error
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Errors

• Type I error consists of rejecting the null
  hypothesis when H0 is actually true.
• Type II error Researcher fails to reject a null
  hypothesis that is really false.

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Alpha level

• Level of significance is a probability value
  that defines the very unlikely sample outcomes
  when the null hypothesis is true.
• Whenever an experiment produces very unlikely
  data, we will reject the null hypothesis.
• The Alpha level defines the probability of Type I
  error.

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Critical region

• It is composed of extreme sample values that are
  very unlikely to be obtained if the null
  hypothesis is true.

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Significance

• A psychologist develops a new inventory to
  measure depression. Using a very large
  standardization group of normal individuals, the
  mean score on this test is µ55 with s12 and the
  scores are normally distributed. To determine if
  the test is sensitive in detecting those
  individuals that are severely depressed, a random
  sample of patients who are described depressed by
  a threapist is selected and given the test.
  Presumably, the higher the score on the inventory
  is, the more depressed the patient is. The data
  are as follows 59, 60, 60, 67, 65, 90, 89, 73,
  74, 81, 71, 71, 83, 83, 88, 83, 84, 86, 85, 78,
  79. Do patients score significantly different on
  this test? Test with the .01 level of
  significance for two tails?