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9-1 Statistical Tests
Is the global temperature increasing
Did the laws requiring hands free cell phone
use result in a decrease in auto accidents?
Is the housing market truly “cooling off”?
These are all hypotheses.. Or suppositions.
Hypothesis (cont)
A Hypothesis proposes a model; if the
data is consistent with the model, then
there is no reason to think that the
hypothesis is false. If the facts instead
do not fit with the model, then perhaps
the model might be wrong…
The question is, how far off is
Hypothesis (cont)
Hypotheses don’t involve estimating
population parameters but instead are
about the reasonableness of the value
of the parameter.
Null Hypotheses = Ho
Parameter is correct as stated
Alternative Hypotheses = H1 = HA
Parameter is not correct as stated
Hypothesis (cont)
A Toyota salesman tells you that the Prius
gets 45 mpg. You think he is
Null Hypotheses = Ho : μ = .75
Alt. Hypotheses = H1(HA ) : μ < .75
Hypothesis (cont)
An allergy drug has been tested and the
claim is that 75% of patients in a large
clinical trial find their symptoms
significantly reduced.
Null Hypotheses = Ho : p = po= .75
Alt. Hypotheses = H1(HA ) : p ≠ .75
Hypothesis (cont)
Notice that the null is always an equal
statement, while the alternate will be a
greater than (right tailed), less than
(left tailed) or not equal to (two tailed).
This is important for reading the charts;
the calculator will read the alternate
hypothesis correctly without you saying
left, right etc.
How are we going to decide whether to
accept the null hypothesis or reject?
That is, what are we really doing?
How far off is data from the assumed
statistic? What is the probability that
the observed data is realistic with the
assumed statistic?
Test Statistic
x μ
With normal x and known σ, what is the
probability that the  value (or z
value) exists with the assumed μ
Four Steps
1. State the null hypothesis as well as the
2. Check the model (normal)
3. Calculate the test statistic.
The goal is to get the P value (the probability that the observed
statistic value could occur if the null hypothesis is correct). The
smaller the P-value, the more likely the rejection of Ho. It
suggests that results are less likely due to chance.
4. State the conclusion. Either reject or fail to
reject the null hypothesis.
Get in the habit of again drawing pictures to
visualize the test (left, right, two)
Mean Example
Rosie, an aging sheep dog in Montana gets regular
checkups from the local vet. Let x be a random
variable that represents Rosie’s resting heart rate
(beats per min). From past experience, the vet
knows that x has a normal distribution with σ = 12
and, for dogs of this breed, μ = 115.
Over the past six weeks, Rosie’s heart rate measured
an average of 105.0 (six different measurements.
The vet is concerned that Rosie’s heart rate may be
slowing. Do the data indicate that this is the case?
Step 1: Ho : μ = 115 HA : μ < 115
Step 2: Independence? Likely. Randomization?
indicates anything to the contrary.
Step 3: find z -2.04
Step 4: As P( < 105.0) = P(z < -2.04)= 0.0207.
That is, the probability of getting a sample
mean below 105.0 is less than 2%, so reject
Ho and accept HA.
Note: we have NOT proved that the alternate is true. *
Types of Errors
Type 1: Rejecting the null hypothesis
when it is actually true
(false positive – diagnosing a healthy person with a
disease, convicting an innocent person)
Type 2: Accepting the null hypothesis
when it is false.
(false negative – diagnosing a sick person as free from
disease, allowing a guilty person to go free)
Levels of Significance
α (alpha) = probability of rejecting Ho
when it is true
i.e. probability of a Type 1 error
β (beta) = probability of accepting Ho
when it is false
i.e. probability of a Type 2 error
Obviously we want α and β to be as
small as possible
Levels of significance (cont)
The true situation
What the
Reject Ho
Accept Ho
Ho true
Ho false
Type 1 = α
Type 2 = β
The power of the test is its ability to detect a false hypothesis.
Power of the test = 1 – β
The lower value for β, the more stringent the test.
The power of the statistical test will increase as α increases,
but a larger value of α increases the likelihood of a type 1
Level of Significance (cont)
Typically α is decided in advance. Then
the P-value is determined.
P-value ≤ α then reject the null
hypothesis and say that the data is
statistically significant at the given
P-value ≥ α then do not reject the null
hypothesis. *
The price to earnings ratio is an important tool in
financial work. A random sample of 14 large US
banks gave the following P/E ratios (source: Forbes)
The sample mean is approximately 17.1. Generally
speaking, a low P/E ratio indicates a “value” stock.
A recent copy of the Wall Street Journal indicated
that the P/E ratio of the entire S&P 500 stock index is
μ = 19. Let x b e a random variable representing
the P/E ratio of all large U.S. bank stocks. We
assume that x has a normal distribution and a σ =
4.5. Do these data indicate that the P/E ratio of all
U.S. bank stocks is less than 19? Use α = 0.05.