MATH 2400
Ch. 15 Notes
Two Types of Statistical Inference
1) Confidence Intervals
•
Used when your goal is to estimate a population
parameter
2) Tests of Significance
•
Used to assess the evidence provided by data
about some claim concerning a population
Example of a Test of Significance
Jordan claims that he makes 75% of his free
throws. You tell him to “prove it.” He takes 20
shots from the free throw line and makes 8 of
the 20. You conclude that he was lying.
What is the probability that he was telling the
truth and still makes 8 out the 20 free throws?
Test of Significance, the Basics
• Based on asking what would happen if we
repeated the sample or experiment many
times.
• Assume perfectly random SRS from an exactly
Normal population.
• Assume we know σ.
Example 1
Artificial sweeteners lose their sweetness over time.
In a study, trained testers sipped cola along with
drinks of standard sweetness and scored the cola on
a “sweetness score” of 1 to 10. The cola was stored
for a month at high temperature to simulate storing
for 4 months at room temperature. Each taster
scored the cola again. This is a matched pairs
experiment. Our data represents the difference
(score before – score after) in the tasters’ scores.
The larger the score, the larger the loss of
sweetness.
Example 1 continued…
2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3
Most are positive, that is, most tasters found a
loss of sweetness. But, the losses are small.
Two tasters (the negative scores) though the
cola gained sweetness. The average
sweetness lost is given by the sample mean
x=̄ 1.02. Is this data good evidence that the
cola lost sweetness in storage?
Example 1 continued…
The reasoning is the same as the free throw
example. We make a claim and ask if the data
gives evidence against it. We seek evidence
that there is a sweetness loss, so the claim we
test is that there is not a loss. In that case,
μ=0.
We can calculate the
standard deviation…
Example 1 continued…
Let’s consider…
1) Our case where x̄
= 1.02 and
2) Another case
where some cola
already on the
market was
sampled, and had
a mean loss of x̄ =
0.3.
Example 1 continued…
For the situation, we assume x̄ = 0 and we just
calculated the margin of error to be .316, so
Our case: x̄ = 1.02
Another case: x̄ = 0.3
x̄ margin of error = 0 .316
Since Our case, falls well outside of this range,
we can say that the evidence shows that the
cola did lose sweetness.
Since the other case falls within this range, we
can not say that the cola lost sweetness.
Stating Hypotheses
We start with a careful statement of the claims we
want to compare. We look for evidence against a
claim, so we start with the claim we seek evidence
against, such as “no loss of sweetness.”
Null & Alternate Hypotheses
H0 will represent the Null Hypothesis
Ha will represent the Alternate Hypothesis
Hypotheses always refer to a population, not to
a particular outcome. Be sure H0 and Ha are in
terms of population parameters.
For Example 1: H0: μ = 0
Ha: μ > 0
Example 2
Does the job satisfaction of assembly workers
differ when their work is machine paced
rather than self-paced. Assign workers either
to an assembly line moving at a fixed pace, or
to a self-paced setting. All subjects work in
both settings, in random order. This is a
matched pairs design. After two weeks, the
workers take a test of job satisfaction. The
response variable is the difference in
satisfaction scores, self-paced minus machinepaced.
Example 2 continued.
So, we are trying to determine if there is a
difference. So, our Null Hypothesis will be that
there is no difference.
H0: μ = 0
Ha: μ ≠ 0
The Hypotheses should be expressed before looking
at the data. It is tempting to look at the data and
frame the hypotheses to fit what the data shows.
The data in this example showed that workers
were more satisfied with self-paced work, which
should not influence Ha.
Example 3
State the Null and Alternate Hypotheses
Example 4
State the Null and Alternate Hypotheses
Example 5
Explaining the Null Hypothesis
Writing the Null Hypothesis in a way in which we
want to find evidence against may seem weird
at first. Think about it like a criminal trial. The
defendant is “innocent until proven guilty.”
That is, the Null is innocent and the
prosecution must try to provide convincing
evidence against this hypothesis.
We are trying to prove the Null false.
P-Value and Statistical Significance
Small P-values are evidence against H0 because
they say that the observed result would be
unlikely to occur if H0 was true.
Large P-values fail to give evidence against H0.
Example 1 Revisited
The study of sweetness lost tests the hypotheses
H0: μ = 0
Ha: μ > 0
Because Ha states that μ > 0, values of x̄ greater
than 0 favor Ha over H0. To calculate the pvalue…
 p = P(x̄ > 0.3) = .1711
(by looking at z-table)
This is not strong evidence against H0.
Example 1 Revisited…
However, for our data which gave x̄ = 1.02…
 p = P(x̄ > 0.3) = .0006
(by looking at z-table)
This is strong evidence against H0 and in favor of
Ha.
Example 2 Revisited…
Consider the job satisfaction example…
H0: μ = 0
Ha: μ ≠ 0
Suppose we know that the job satisfaction
scores follow the Normal Distribution with
σ=60. Data from 18 workers give x̄ = 17.
Which means they prefer self-paced on the
average. Because the alternative is two-sided.
The p-value is the probability of getting an x̄ at
least as far from μ=0 in either direction as the
observed x̄ = 17.
Example 2 Revisited…
So, with σ = 60, n = 18, we get
Which means,
and, by looking at the z-table…
p = 2P(-17< x̄ < 17) = 2(.1151) = .2302.
This is not strong evidence against H0.
P-Values…
We have said that a p-value of…
• 0.0006 was strong evidence against H0.
• 0.1711 and .2302 were not strong evidence
against H0.
There is “no rule” for how small a p-value has to
be to reject H0, it’s a matter of judgment.
P-Values
However, there are some fixed values that are in
common use as standards for evidence against
H0.
p = 0.05 means that the probability of this
happening is 5% when repeated many times
p = 0.01 means that the probability of this
happening is 1% when repeated many times
These are called significance levels (α).
Tests for Significance
z Test for a Population Mean
Draw an SRS of size n from a Normal population
that has unknown mean μ and known
standard deviation σ. To test the null
hypothesis that μ has a specified value,
H0: μ = μ0
Calculate the one-sample z test statistic
z Test for Population mean
Example 6
The systolic blood pressure for males 35 to 44
years of age has mean 128 and st. dev. 15. A
large company looks at the medical records of
72 randomly chosen workers in this age group
and finds that the mean systolic blood
pressure is x̄ = 126.07. Is this evidence that
the company’s executives have a different
mean systolic blood pressure from the general
population?
Example 6 continued…
The Null Hypothesis will be no difference and our
Alternate Hypothesis will be two-sided because x̄
could be above or below μ.
H0: μ = 128
Ha: μ ≠ 128
To find the p-value, look for z < -1.09. Our table
shows P(z < -1.09) = 0.1379. So,
p = 2P(z < -1.09) = (2)(0.1379) = 0.2758
This is not strong evidence that these workers’ blood
pressures differ from the rest of the population.
Before Doing a Z Test…
1) Verify SRS
2) Normal Distribution
3) Know σ
Example 7
Here are 6 measurements of the electrical
conductivity of an iron rod:
10.08 9.89 10.05 10.16 10.21 10.11
The iron rod is supposed to have conductivity 10.1.
Do the measurements give good evidence that the
true conductivity is not 10.1? The 6 measurements
are an SRS from a population with a Normal
distribution with σ = 0.1.
Ti-84 Z Tests
STAT, TESTS tab, 1: Z-Test…
This is what our screen should look like for
Example 1.
After hitting “Calculate”
Example 1 Information (1 sided)
Set 1
μ0 = 0
σ= 1
x̄ = 1.02
n = 10
p=
Set 2
μ0 = 0
σ= 1
x̄ = 0.3
n = 10
p=
Example 2 Information (2 sided)
μ0 = 0
σ = 60
x̄ = 17
n = 18
p=
Example 6 Information (2 sided)
μ0 = 128
σ = 15
x̄ = 126.07
n = 72
p=
Example 7 Information (2 sided)