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PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 7
Continuous Distributions
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Suppose that a school band….
1
1 
  90   80   70   60    70   60   50   50   40   30    40   30   20   

                                          
100  
  15   15   15   15    15   15   15   15   15   15    15   15   15   



 15 
One class is
not included
Two classes are
not included
Three
classes are
not included
Or
consisting
of only one
class
Hints
n
Pr(
i 1
n
Ai )   Pr( Ai 
i 1
n 1
Ai )
i 1
These are disjoint
Corrections
Find the probability of the subset of points
such that
y  1  x2
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.
Question
If A and B are independent events and Pr(B)<1,
Pr( Ac B c )  ?
Question
Suppose that a random variable X has discrete
distribution with the following probability function:
c
 2 for x  1, 2,..
f ( x)   x

otherwise
0
Find the value of the constant
The probability density function (p.d.f.)
Every p.d.f f must satisfy the following
requirements f ( x )  P ( X  x )
two
f ( x)  0, for all x,
+
 f(x)dx=1
-
Ex: Suppose that X has a binomial with n=2 and p=1/2.
Find f(x) and Pr( X  1,5)
Example
EX: Suppose that the p.d.f of a certain random variable
X is as follows:
cx 2 for 1  x  2
f ( x)  
 0 otherwise
Find the value of a constant c and sketch the p.d.f.
Find the value of Pr( X  3 )
2
Sketch probability distribution function
Normal p.d.f.
f ( x) 
e
 ( x   )2 / 2 2
 2
,   0,     ,   x  
Example
EX:Let we have a normal distribution with
mean 0 and variance 1.
Find
P(0  X  2)
P(2  X  2)
P(0  X  1,53)
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?
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.
The distribution of sample means
EX: population: 1, 3, 5, 7
a.
Sample size: 2,
b. Pr( X  4)  ?
The standard error of X
The standard deviation of the distribution of
sample means is called the standard error
of X
1.
2.
The standard deviation of the population
The sample size

standard error   X 
n
Example
A population of scores is normal, with µ=50 and σ=12.
Describe the distribution of sample means for
samples size n=16 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
1. The population from which the samples are selected is normal
distribution.
2. The number scores (n) in each sample is relatively large, around
30 or more.
Example
EX: A skewed distribution has µ=60 and σ=8.
a. What is the probability of obtaining a
sample mean greater than X =62 for a
sample of n=4?
b. What is the probability of obtaining a
sample mean greater than X =62 for a
sample of n=64?
Introduction to hypothesis
testing
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.
Hypothesis testing (Steps)
1.
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.
H1:predicts that IV will have an effect on
the DV for the population
Hypothesis testing
2.
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.
Hypothesis testing
Collecting sample data
4.
Evaluating the null hypothesis
The researcher compares the data X 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
3.
Errors in hypothesis testing
ACTUAL SITUATION
Researcher
decision
No effect,
H0 True
Effect Exists,
H0 False
Reject H0
Type I error
Decision correct
Retain H0
Decision
correct
Type II error
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.
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.
Critical region
It is composed of extreme sample values that are
very unlikely to be obtained if the null
hypothesis is true.
  .05,
  .01,
z  1,96
z  2,58
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 µ=55
with σ=12 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?