Transcript Document

Hypothesis Test II: t tests
Test
for one population mean
Test for the difference of two
population means
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Hypothesis Tests for Population
Parameters
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
Population proportion

Population mean

Difference between two population means
5 Steps in Hypothesis Tests
Step 1. Determine the null (Ho) and
alternative (Ha) hypotheses
Ho : the population parameter = a constant
(called the null value)
Ha : upper or lower or two-tailed
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5 Steps in Hypothesis Tests

Step 2. Find an appropriate test statistic and
pre-set the level of significance (called a
level)
Test statistic =
(Sample estimate – null value)
(S.E. of sample estimate@ null value)
a= 0.05 (or 0.01 or 0.10)
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5 Steps in Hypothesis Tests

Step 3. Check the conditions and find the pvalue assuming Ho is true
The calculation of the p-value depends on the
Ha.
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5 Steps in Hypothesis Tests
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Step 4. Decide whether or not reject Ho
based on the p-value
p-value > a
p-value < a
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

fail to reject Ho
reject Ho (= accept Ha)
5 Steps in Hypothesis Tests
Step 5. Report the result in the context of the
situation
Reject Ho 
There is sufficient evidence to support (Ha).
Fail to reject Ho 
There is not sufficient evidence to support (Ha).
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Hypothesis Test for a Population Mean

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Step 1. Ho: m = mo
Three possible Ha’s:
Ha: m = mo (two-tailed)
or
Ha: m < mo (lower-tailed)
or
Ha: m > mo (upper-tailed)
Hypothesis Test for a Population Mean

Step 2. test statistic
x  mo
t=
s/ n
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Hypothesis Test for a Population Mean
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1.
2.
Step 3:
Condition: Must be a random sample and
the data must be bell-shaped (or normal)
When the above conditions are met, use tTable to find p-value (the required df = n-1)
This test is called “one-sample t-test.”
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Examples
The average cost of 30 randomly selected used
cars in a used car lot was $5,000 with
sample standard deviation of $1,000.
 Find the p-value for testing
Ho: m = $4,500 vs. Ha: m = $4,500.
Answer: t = 2.74 and 0.006 < p-value < 0.016
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Examples

Find the p-value for testing
Ho: m = $4,500 vs. Ha: m > $4,500.

Find the p-value for testing
Ho: m = $4,500 vs. Ha: m < $4,500.
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Hypothesis Test for the Difference of
Two Population Means
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Step 1. Ho: m1 = m2
Three possible Ha’s:
Ha: m1 = m2 (two-tailed)
or
Ha: m1 < m2 (lower-tailed)
or
Ha: m1 > m2 (upper-tailed)
Hypothesis Test for the Difference
between Two Population Means

Step 2. test statistic
t=
x1  x2
2
1
2
2
s
s

n1 n2
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Hypothesis Test for the Difference
between Two Population Means
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1.
2.
Step 3:
Must be two independent random samples; the both
samples must be bell-shaped (or normal)
When the above conditions are met, use t-Table to
find the p-value (the required df is the smaller one of
(n1-1) and (n2-1))
This test is called “Two-sample t-test.”
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Examples
The average waiting time for 50 students to
connect to the college server from the
dormitories was 5 seconds, while the average
time to connect for 20 students living offcampus apartments was 10 seconds.
Assume the times to connect in the
dormitories and the apartments are both bellshaped. The sample standard deviations
were 2 seconds for dormitories and 8
seconds for apartments
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Examples
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
Find the p-value for testing
Ho: m1 = m2 vs. Ha: m1 = m2

Find the p-value for testing
Ho: m1 = m2 vs. Ha: m1 < m2
Answer: t = -2.76
1: dormitories; 2: apartments