Download Hypothesis Testing Example

Populations and Samples
Hypothesis Testing Example
Lecture Objectives
You should be able to:
1. Set up the Null and Alternate
Hypotheses
2. Conduct a one-sample Hypothesis Test
for the Mean.
3. Interpret the results
The Situation
You believe that the current price of
unleaded regular gasoline is less than
$4.00 on average nationwide, and
wish to prove it.
Set up the hypothesis and test it.
Null and Alternate Hypotheses
What we wish to prove is called the Alternate
Hypothesis. The opposite of that is the Null,
which must be assumed and shown to be
unlikely, based on sample data.
H0: μ = 4.00
Ha: μ < 4.00
What constitutes proof?
Any conclusion based on a sample may be
wrong. What probability (at most) of being
wrong is acceptable to you?
 (alpha), or the acceptable
This is called
Type I Error.
Let
 = 0.05 (or 5%)
The Sample Data
A sample of 49 gas stations nationwide
shows average price of unleaded is $ 3.87
and a standard deviation of $ 0.15 .
Could this sample have come from a population
where the Mean was in fact $4.00 (or greater)?
Assume the null is true, and this sample did
in fact come from such a population.
Sampling Distribution if H0 True
What would the distribution of sample means from such a
population look like? From the Central Limit Theorem, we
have the following:
x  
x
=
s
= $4.00
n = 0.15/√49 = $ 0.02143
The Test Statistic
How far from the assumed mean of 4.00 is the observed
sample mean of 3.87?
Measured in Standard Errors, this is the t-statistic.
t = (3.87- 4.00)/0.02143 = -6.06
How likely is it that a sample mean would be this
far away (or farther) from the population mean?
p-value
The probability that a value would be as extreme
as (or more extreme than) 6.06 SEs below the
Mean is:
0.0000001!
[In Excel, =TDIST(6.06,48,1)]
This is called the p-value of the Hypothesis test.
Conclusion
If the null were true (the average price were in
fact 4.00), there is only a 0.0000001
probability that you would pick a sample with
a mean of 3.87 or smaller from such a
population.
Still, you did pick it!
Therefore, either the null must be false (and
therefore you proved your case) or you picked
an extremely rare sample.
Conclusion (2)
You can conclude that the sample could not have
come from a population with Mean = 4.00 as
assumed, and instead must have come from one with
Mean < 4.00.
The chance that you are wrong is less than 5%, your
tolerance level.

In other words, p < , hence you proved the case
beyond reasonable doubt.