Transcript No Slide Title

EXAM REVIEW
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson
Addison-Wesley
MODELING THE DISTRIBUTION OF
SAMPLE PROPORTIONS (CONT.)

A picture of what we just discussed is as follows:
EXAMPLE



A candy company claims that its jelly bean mix
contains 45% red jelly beans.
Suppose that the candies are packaged at
random with 100 jelly beans each.
What’s the probability that a bag will contain
more than 50% red jelly beans?
Slide
1- 3
PROPORTIONS: ONE-SAMPLE - SE AND Z*
HYPOTHESIS TESTING

The conditions for the one-proportion z-test are the
same as for the one proportion z-interval. We test the
hypothesis
H 0: p = p 0
using the statistic
pˆ  p0 

z
SD  pˆ 
where

SD  pˆ  
p0 q0
n
When the conditions are met and the null hypothesis is
true, this statistic follows the standard Normal model,
so we can use that model to obtain a P-value.
Slide
1- 4
PROPORTIONS: ONE-SAMPLE CONFIDENCE INTERVALS
When the conditions are met, we are ready to find the
confidence interval for the population proportion, p.
 The confidence interval is

pˆ  z  SE  pˆ 

where

ˆˆ
SE( pˆ )  pq
n
The critical value, z*, depends on the particular
confidence level, C, that you specify.
Slide
1- 5
PROPORTIONS: ONE-SAMPLE - EXAMPLE
A state university wants to increase its retention
rate of 10% for graduating students from the
previous year.
 After implementing several new programs to
increase retention during the last two years, the
university re-evaluated its retention rate using a
random sample of 352 students.
 The new retention rate was 12%.

Slide
1- 6
PROPORTIONS: ONE-SAMPLE - EXAMPLE
Test the hypothesis that the retention rate had
increased and state your conclusion with a 98%
confidence interval.
 Also test the hypothesis with a z-test using a
significance level of 0.01

Slide
1- 7
MEANS: ONE-SAMPLE – T-TESTING
A practical sampling distribution model for
means
When the conditions are met, the standardized sample
mean
y 
t
SE  y 
follows a Student’s t-model with n – 1 degrees of
freedom.
s
We estimate the standard error with SE  y  
n
Slide
1- 8
MEANS: ONE-SAMPLE - HYPOTHESIS

One-sided alternatives
Ha: μ>hypothesized value
 Ha: μ <hypothesized value


Two-sided alternatives

Ha: μ ≠ hypothesized value
Slide
1- 9
MEANS: ONE-SAMPLE – CONFIDENCE INTERVALS
One-sample t-interval for the mean

When the conditions are met, we are ready to find the
confidence interval for the population mean, μ.

The confidence interval is
 SE  y 

n1
where the standard error of the mean is
y t
s
SE  y  
n
*
 The critical value tn1depends on the particular
confidence level, C, that you specify and on the number
of degrees of freedom, n – 1, which we get from the
sample size.
Slide
1- 10
MEANS: ONE-SAMPLE – EXAMPLE
A sociologist develops a test to measure attitudes
about public transportation, and 50 randomly
selected subjects are given the test.
 Their mean score is 85 and their standard
deviation is 15.
 Construct a 95% confidence interval for the mean
score of all such subjects.

Slide
1- 11
PROPORTIONS: TWO-SAMPLE - HYPOTHESIS


The typical hypothesis test for the difference in
two proportions is the one of no difference. In
symbols, H0: p1 – p2 = 0.
The alternatives:



Ha: p1 –p2 > 0
Ha: p1 –p2 < 0
Ha: p1 –p2 ≠ 0
Slide
1- 12
PROPORTIONS: TWO-SAMPLE - SE AND Z*
HYPOTHESIS TESTING
 We use the pooled value to estimate the standard error:
pˆ pooled qˆ pooled pˆ pooled qˆ pooled
SE pooled  pˆ1  pˆ 2  

n1
n2

Now we find the test statistic:
pˆ1  pˆ 2   0

z
SE pooled  pˆ1  pˆ 2 

When the conditions are met and the null hypothesis is
true, this statistic follows the standard Normal model,
so we can use that model to obtain a P-value.
Slide
1- 13
CALCULATING THE POOLED PROPORTION

The pooled proportion is
pˆ pooled
where

Success1  Success2

n1  n2
Success1  n1 pˆ1
and
Success2  n2 pˆ 2
If the numbers of successes are not whole numbers, round
them first. (This is the only time you should round values in
the middle of a calculation.)
Slide
1- 14
PROPORTIONS: TWO-SAMPLE CONFIDENCE INTERVALS


When the conditions are met, we are ready to find the
confidence interval for the difference of two proportions:
The confidence interval is
 pˆ1  pˆ 2   z

 SE  pˆ1  pˆ 2 
where
SE  pˆ1  pˆ 2  

pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
The critical value z* depends on the particular confidence
level, C, that you specify.
Slide
1- 15
PROPORTIONS: TWO-SAMPLE - EXAMPLE
A survey of randomly selected college students
found that 50 of the 100 freshman and 60 of the
125 sophomores surveyed had purchased used
textbooks in the past year.
 Test for a difference between the two student
groups using a significance level of 0.05.

Slide
1- 16
MEANS: TWO-SAMPLE - HYPOTHESIS

One-sided alternatives
Ha: μ1 – μ2 >0
 Ha: μ1 – μ2 <0


Two-sided alternatives

Ha: μ1 – μ2 ≠ 0
Slide
1- 17
MEANS: TWO-SAMPLE – T-TESTING

When the conditions are met, the standardized sample
difference between the means of two independent groups
y1  y2    1  2 

t
SE  y1  y2 

can be modeled by a Student’s t-model with a number of
degrees of freedom found with a special formula.
We estimate the standard error with
SE  y1  y2  
s12 s22

n1 n2
MEANS: TWO-SAMPLE – DEGREES OF
FREEDOM

The special formula for the degrees of freedom for
our t critical value is a bear:
2
 s12 s22 
  
 n1 n2 
df 
2
2
1  s12 
1  s22 
  
 
n1  1  n1  n2  1  n2 

Because of this, we will let technology calculate
degrees of freedom for us!
Slide
1- 19
MEANS: TWO-SAMPLE – CONFIDENCE INTERVAL
When the conditions are met, we are ready to find the confidence
interval for the difference between means of two independent
groups.
The confidence interval is
 y1  y2   t

df
 SE  y1  y2 
where the standard error of the difference of the means is
SE  y1  y2  
s12 s22

n1 n2
The critical value depends on the particular confidence level, C, that you
specify and on the number of degrees of freedom, which we get from the
Slide
sample sizes and a special formula.
1- 20
MEANS: TWO-SAMPLE – EXAMPLE
Two types of cereal brands are being tested for
sugar content
 Brand Yummy – n=100, Ӯ=5, s=2
 Brand Yuck – n=150, Ӯ=4.5, s=2


Construct a 95% confidence interval for the
difference between the two brands.
Slide
1- 21
UPCOMING IN CLASS
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Exam #2 Wednesday
Data Project Due by 5pm Thursday December 5th
via email or my department mailbox.
Finals (optional)
Wednesday December 11th
 1-3pm
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