9.1 – The Basics
Ch 9 – Testing a Claim
Jack’s a candidate for mayor against 1 other person,
so he must gain at least 50% of the votes. Based on
a poll of voters just before the election, can Jack be
confident of victory?
Random
Selection of
a chip.
White chip =
Voting for
Jack
Construct
Scatter plot:
X-axis = # of
chips
Y-axis =
proportion of
white
Keep drawing
until your
confident
Jack will win
We will do this again with a different bag with a
different proportion of white chips.
Two important Aspect of Hypothesis Testing:
1. increasing n help us to be more confident
that there is a difference between what is
observed and what is claimed.
2. It is easier to reject what is claimed if the true
parameter is farther away from what is
hypothesized
Goal of the Two Inferences
Confidence Intervals Estimate a population parameter
Significance Test Assess the evidence provided by data about some claim
concerning a population
Should Jack be
confident in a
victory?
Suppose a random sample of 100
voters shows that 56 will vote for Jack.
Is his confidence warranted? Maybe
so, but there are two explanations of
why the majority of voters in the
sample seem to favor Jack.
Reasons:
1. In reality less than 50%
favor Jack, and the
sample result was just do
to sampling variability. If
this is true, then Jack
shouldn’t be confident of
a victory.
2. The sample is above 50%
because more than 50%
will actually vote for Jack.
Then he should believe
this sample and be
confident of victory ONLY
IF he can rule out the first
reason!
A Fathom Simulation was done to simulate 400 samples
of size 100 from the population in which exactly 50% of
the voters support Jack. 55 of the 400 trials had a sample
proportion of 56% or higher.
Jack should be worried of winning!
13.75% is a high chance that these
results would have happened by
pure chance alone if the true value
is 50%. It is plausible that 50% of
the voters do not favor Jack.
Stating our Hypotheses
Null Hypothesis – H0 : this is the claim we are testing.
This is typically a statement of no difference.
H0: (parameter symbol) = ##
Alternate Hypothesis – Ha : this is the claim about the
population that we are trying to find evidence for
This statement is one-sided or two-sided.
Ha: (parameter symbol) > ##
Ha: (parameter symbol) < ##
Ha: (parameter symbol) = ##
ALWAYS STATE THE HYPOTHESIS BEFORE USING
THE DATA!!!!
HYPOTHESES ARE ABOUT A POPULATION.
YOUR EVIDENCE IS YOUR SAMPLE.
P-VALUE
»Assuming the Ho is true, the probability the p-hat of x-
bar would take on a value more extreme than the 1
observed
»Smaller = the more rare our sample results are and the
more evidence we have to support Ha and against Ho
»Larger = the more likely our sample results our and the
less evidence we have to support Ha and we fail to
reject Ho.
Interpret the P-value for Jack’s chances of
winning
»If exactly 50% support Jack, there is a 13.75% chance in
a sample of 100, 56% or more would actually support
him by random chance alone.
»Since this is not a rare occurrence, Jack should be
worried that 50% actually support him.
a) Describe the parameter of interest.
b) State the hypotheses for performing a significance test.
a) Interpret the P-value.
b) Does the data provide convincing evidence against Ho?
Explain
How low is low?
»Statistically Significant level – alpha
»Stated BEFORE the test is conducted.
»Typically 0.05 (or 0.01) it depends on the context and
severity of the results.