6. Determine if the data support the theory
More details about hypotheses
 A priori,
 Null must be falsifiable.
 Null should give simplest possible explanation – e.g. the
observed difference is because of sampling error (normal
variation that exists for a random variable).
 Alternative is the next simplest – the observed difference is
because of something other than randomness.
 Hypotheses are about parameters ( or p or ρ), the null
contains the equal sign; the alternative contains the
inequalities.
Complete and submit In-Class Activity – Writing Hypotheses
Page 269
The reasoning process for deciding which hypothesis the data
supports is the same for any parameter (p or  or ρ).
1. Assume the null hypothesis is true.
2. Gather data and calculate the statistic.
3. Determine the likelihood of selecting the data that produced the
statistic or could produce a more extreme statistic.
4. If the data are likely, they support the null hypothesis.
However, if they are unlikely, they support the alternative
hypothesis.
Likely and unlikely statistics are found through theoretical
sampling distributions
Suppose your hypotheses were:
H0: p = 0.25
H1: p > 0.25
Assuming the null hypothesis is true, does it seem likely or
unlikely that the value, or more extreme values, would be selected
if the sample proportion is: 0.24? 0.28? 0.30? 0.36? 0.40?
The direction of extreme is the direction on the number line that
will lead to selecting the alternative.
Sampling distribution if p = 0.25.
Notice the distribution and the center of the distribution.
.
Sampling distribution if p = 0.5
Sampling distribution if p = 0.6
In-class Activity Sampling Distributions Page 271
Compare all three sampling distributions when they are on the
same graph.
Suppose our question is: Does a majority of people in the US
believe drilling for oil in the Arctic Ocean should be allowed?
If our hypotheses were
H0: p = 0.5
H1: p > 0.5
If we represent the alternative hypothesis with the p = 0.6
distribution, from which distribution are we most likely to get a
pˆ  0.52 or higher value?
What about
pˆ  0.58 or
higher value?
Now consider the following in which the alternative is based on
p = 0.55.
What if pˆ  0.52 .
We can make errors without doing anything wrong!
Type I error - the data supports the alternative hypothesis
when the null hypothesis is true.
Type II errors –the data supports the null hypothesis when
the alternative is true.
Probability is our tool for understanding errors and making
decisions.
We want to know the probability of making these errors.
Discuss probability briefly.
Probability is always a fraction or a decimal between 0 and 1. This
is shown generically as 0  P(A)  1 where P(A) represents the
probability of event A.
Probability of a Type I error is α, called the level of significance.
The probability of a Type II error is β. The probability of correctly
selecting the alternative is called power (1 – β).
The Evidence
upon which
the decision is
based.
The Data
Supports H0
The Data
Supports H1
The True Hypothesis
H0 Is True
H1 Is True
No Error
Type II Error
Probability: β
Type I Error
No Error
Probability: α Probability:
Power
Because some values clearly support the null hypothesis,
others clearly support the alternative hypothesis but some
do not clearly support either, then a decision has to be
made, before data is ever collected (a priori), as to the
probability of making a Type I error that is acceptable to
the researcher.
In our current example, if a level of significance of 0.05 is
used, a decision rule line is placed on the distribution that
separates the likely, from the 5% of unlikely values,
dependent upon the direction of the extreme.
β
5% of the null values are higher than 0.54 so there is a 5%
chance we would make a Type I error.
19% of the values of the alternative distribution are below
0.54, so there is a 19% chance of making a Type II error.
Power is 81%.
0.54 is called the critical value.
How to Make Statistical Decisions (p-value, alpha, significant)
The choice of which hypothesis is to be supported must be
based on evidence. The evidence is provided by statistics
calculated from the selected data.
The probability of getting the statistic, or more extreme statistics
assuming the null hypothesis is true, is found. This probability is
called a p-value.
If the p-value is less than or equal to α, the data supports the
alternative hypothesis. If the p-value is greater than α, the data
supports the null hypothesis.
When the data supports the alternative hypothesis, the data are said
to be significant. When the data supports the null hypothesis, the
data are not significant.
Demonstrate p-value if
pˆ  0.55 .
Then if
pˆ  0.52 .
In-Class Activity (p-value and alpha) page 272
Conclusions
Conclusions in this course (book and exams) will be expressed in a
way that is similar to that used by researchers who publish in
scholarly journals.
If the data was = 0.55, we write: At the 0.05 level of
significance, the proportion of the population that believes drilling
in the Arctic Ocean should be allowed is significantly greater than
0.50 (p = 0.02, n = 500).
If the data was = 0.52, we write: At the 0.05 level of
significance, the proportion of the population that believes drilling
in the Arctic Ocean should be allowed is not significantly greater
than 0.50 (p = 0.22, n = 500).
Level of Significance
The most commonly used values for α are 0.05, 0.01 and 0.10.
Draw a number line showing alphas and p-values and the zones of
significance.
α:
0 0.01 0.05
0.10
1
The choice for a level of significance should be based on several
factors.
1. If the power of the test is low because of small sample
sizes or weak experimental design, a larger level of significance
should be used.
2. Keep in mind the ultimate objective of research – “to
understand which hypotheses about the universe are correct.
Ultimately these are yes and no decisions.” (Scheiner, 2001) p8.
Statistical tests should lead to one of three results. One result is
that the hypothesis is almost certainly correct. The second result
is that the hypothesis is almost certainly incorrect. The third
result is that further research is justified. P-values within the
interval (0.01,0.10) may warrant continued research, although
these values are as arbitrary as the commonly used levels of
significance.
3. If we are attempting to build a theory, we should use
more liberal (higher) values of α, whereas if we are attempting to
validate a theory, we should use more conservative (lower) values
of α.
The choice of hypothesis that is supported is based on both
alpha and p-value. The researcher determines the former before
data is collected (a priori) and the later is determined from the
evidence. The level of significance cannot be changed if you are
unhappy with the result. This is like “throwing darts against the
wall and then drawing the target around them”.
Notation:
– In the study of statistics there are several words that start with
the letter p and use p as a variable.
– The list of words includes parameters, population,
proportion, sample proportion, probability and p-value.
o The words parameter and population are never
represented with a p.
o Probability is represented with notation that is similar to
function notation you learned in algebra, f(x), which is
read f of x. For probability, we write P(A) which is
read the probability of event A.
o To distinguish between the use of p for proportion and
p for p-value, pay attention to the location of the p.
 When p is used in hypotheses, such as H0: p =
0.5, H1: p > 0.5, it means the proportion of the
population.
 When p is used in the conclusion, such as the
proportion is significantly greater than 0.5 (p =
0.02, n = 500), then it is interpreted as a p-value.
o If the sample proportion is given, it is represented as
= 0.55.
7. Make an evidence-based decision
The meaning of “not significant” - be careful with your words.
Ultimately decisions are stated as being significant (low p-value,
alternative is chosen) or not significant (high p-value, null is
chosen).
 Care must be taken when interpreting the phrase “not
significantly greater than 0.50”.
o This phrasing lets the reader know the null
hypothesis was selected but it does not provide
conclusive evidence that the null hypothesis is
true. It only provides evidence that the alternative
hypothesis is not true.
o The wording can be very tricky. Some authors
are willing to say that the null hypothesis is
accepted. Others say the null hypothesis is not
rejected. Others may say that the alternative
hypothesis is not accepted.
 OLI:
"The data provide enough evidence to reject
claim 1 and accept claim 2"; or
"The data do not provide enough evidence to
reject claim 1."
In particular, note that in the second type of conclusion we
did not say: "I accept claim 1," but only "I don't have enough
evidence to reject claim 1."”
o Claiming that the data support the null hypothesis
means that the data had a high probability of
being selected if the null hypothesis is true.
However, that data could also have been selected
from a population with a parameter that is slightly
different from the one in the null hypothesis.
 For example, a researcher may have the null
hypothesis of H0: p = 0.50 that is not
rejected but the reality could be that the true
proportion is actually p = 0.502.
o By saying we accept the null hypothesis, we
would be saying we really believe the proportion
is 0.500.
o By saying we don’t reject the null hypothesis or
that we support the null hypothesis, we are
concluding that 0.5 is a possible proportion but
acknowledge the true proportion could be some
slightly different value.
o The reality is that if the hypothesis test results are
being used to make a decision, then significant
data leads to the decision that is justified by the
alternative hypothesis while non-significant data
leads to the decision that is justified by the null
hypothesis.
Fracking Video
Example 1 First Attempts, Page 273 - Arsenic
Briefing: Arsenic is a naturally occurring element and also a human produced element
(e.g. fracking, combustion of coal) that can be found in ground water. It causes a
variety of health problems and can lead to death. The EPA limit is 10 ppb, meaning 10
ppb or higher is unsafe.
Problem: A natural gas company wants to put a fracking well on your property. They
will pay you royalties. They promise that the technology they use will not contaminate
the aquifer from which you and others in your community get your water. A year later,
sickness in the community leads health department officials to test your water to
determine if it is contaminated with arsenic. The official will take 5 samples of water
over the next 2 months and decide whether you have safe water or unsafe water based
on the average of these samples. Assume these are the two possible distributions that
exist.
H0: Water is Safe H1: Water is Unsafe
α = 0.30.
What is the direction of the extreme?
Show the decision rule line on both distributions.
What is the critical value?
Label α, β, and power
What is the probability of α?
What is the probability of β?
What is the power?
What is the consequence of a Type I error?
What is the consequence of a Type II error?
Data: What you select from the container that was passed around the classroom
What is the p-value for your data?
Write a concluding sentence:
What decision do you make about your house and water supply?
Example 2: Can a summer job pay for a year’s tuition?
Briefing: There was a time when a student could work a summer job and pay for a
year’s tuition. But that was also a time when the state covered a greater percentage (up
to 90%) of college costs. State funding of higher education has not kept up, so a summer
job is no longer sufficient for paying tuition and students have to rely on loans, which
mean post-graduation debt. The generation that got help with their tuition does not
appear to want to help the next generation.1 Currently, state support is about 30%, but
you think it is less than that.
Is the proportion of tuition paid by tax dollars less than .3 (30%)? If it is, you will
become actively involved in lobbying legislators. If it isn’t, you won’t get involved.
H0: p = 0.30
H1: p < 0.30
α = 0.07.
Show the decision rule line on both distributions.
What is the direction of the extreme?
Label α, β, and power
What is the probability of α?
What is the probability of β?
What is the power?
What is the consequence of a Type I error?
What is the consequence of a Type II error?
Data: A sample of 50 colleges showed that state support was 25%.
Write a concluding sentence:
Based on your decision rule, will you lobby legislators?
1
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