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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Slide 1- 1
Exam Review
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Possible Tests

One-proportion z-test

Two-proportion z-test

One-sample t-test for mean

Two-sample t-test for differences of means
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 3
Inferences about Proportions – One
Sample Test

Everyone (100%) believes in ghosts

More than 10% of the population believes in
ghosts

Less than 2% of the population has been to jail

90% of the population wears contacts
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 4
Proportions: One-sample - Hypothesis

Hypothesis Test
 H0: parameter = hypothesized value
HA: parameter < hypothesized value
 HA: parameter ≠ hypothesized value
 HA: parameter > hypothesized value

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 5
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 6
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
H0: p = p0
using the statistic
where SD  pˆ  

pˆ  p0 

z
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 7
Proportions: One-sample - SE and
hypothesis testing

For a sample proportion, the standard error is
SE  pˆ  

ˆˆ
pq
n
For the sample mean, the standard error is
s
SE  y  
n
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 8
Critical Z* and significance level twosided





α= .20  CI = 80%  z*=1.282
α= .10  CI = 90% z*=1.645
α= .05  CI = 95% z*=1.96
α= .02  CI = 98%z*=2.326
α= .01  CI = 99% z*=2.576
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 9
Critical Values Again (cont.)

Here are the traditional critical values from the
Normal model:

1-sided
2-sided
0.05
1.645
1.96
0.01
2.28
2.575
0.001
3.09
3.29
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 10
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%.
Test an appropriate hypothesis and state your
conclusion.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 11
Comparing Proportions – Two Sample
Test

Women believe in ghosts more than men

People who have been to jail believe in ghosts
more than people who haven’t been to jail

Women smoke more than men

Women use facebook in the bathroom more than
men
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 12
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
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 13
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

where
SE  pˆ1  pˆ 2  

 SE  pˆ1  pˆ 2 
pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
The critical value z* depends on the particular confidence
level, C, that you specify.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 14
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 15
Calculating the Pooled Proportion

The pooled proportion is
pˆ pooled
Success1  Success2

n1  n2
where 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.)
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 16
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.
Construct a 98% confidence interval for the
difference between the two student groups.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 17
Inferences about Means – One Sample
Test

All Priuses have fuel economy > 50 mpg

Ford Focuses get 5 mpg on average

The average starting salary for ISU graduates
>$100,000

The average cholesterol level for a person with
diabetes is 240.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 18
Means: One-sample - Hypothesis


One-sided alternatives
 Ha: μ>hypothesized value
 Ha: μ <hypothesized value
Two-sided alternatives
 Ha: μ ≠ hypothesized value
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 19
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

n 1
where the standard error of the mean is
y t
 SE  y 
s
SE  y  
n

The critical value tn*1 depends 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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 20
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.
We estimate the standard error with SE  y   s
n
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 21
Means: One-sample – Sample Standard
Deviation

The standard deviation, s, is just the square root
of the variance and is measured in the same
units as the original data.
 y  y 
2
s
n 1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 22
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 23
Comparing Means – Two Sample Test

The MPG for the Prius is greater than the MPG
for the Ford Focus

ISU male graduates have a greater starting
salary than women

The cholesterol levels are the same for people
with and without diabetes.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 24
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
s12 s22
SE  y1  y2  

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
sample sizes and a special formula.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 25
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!
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 26
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
s12 s22
SE  y1  y2  

n1 n2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 27
Means: Two-Sample – Standard Error


Remember that, for independent random
quantities, variances add.
So, the standard deviation of the difference
between two sample means is
SD  y1  y2  

 12
n1

 22
n2
We still don’t know the true standard deviations of
the two groups, so we need to estimate and use
the standard error
s12 s22
SE  y1  y2  

n1 n2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 28
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.
Interpret your confidence interval.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 29