Transcript p-value - jmullenkhs

Section 10.2:
Tests of Significance
Hypothesis Testing
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Null and Alternative Hypothesis
P-value
Statistically Significant
Goal of Tests of Significance
To assess the evidence provided
by data about some claim
concerning a population.
Coin Flipping Example
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On a scrap paper record the results of
my coin flips.
Why did you doubt my
truthfulness?
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Because the outcome of the coin
flipping experiment is very unlikely.
How unlikely?
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(.5)^k, where k is the number of flips before you
yelled.
Suppose I claim to consistently make
80% of my volleyball serves.
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Not sure if you believe me, you hand me
a ball and ask me to serve 20 balls.
I make 8 out of those 20 serves.
“Aha,” you say. “Someone who makes
80% of their serves would hardly ever
only make 8 out of 20 shots, so I don’t
believe your claim!”
Significance Test Procedure
STEP 1: Define the population and
parameter of interest. State the null and
alternative hypotheses in words and
symbols.
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Population: My volleyball serves
Parameter of interest: Proportion of serves made
Suppose I am an 80% server. This is a hypothesis and we
think it is false. We will call it the null hypothesis and use the
symbol H0 (pronounced H - nought).
H0 : p = 0.8
You are trying to show that I am worse than an 80% server.
Your alternate hypothesis is:
Ha : p < 0.8
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Significance Test Procedure
STEP 2: Choose the appropriate inference
procedure. Verify the conditions for using
the selected procedure.
We are going to use the Binomial Distribution
Each trial has either a success or a failure
There is a set number of trials
Trials are independent
The probability of a success is constant
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Significance Test Procedure
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STEP 3: Calculate the P-value. The Pvalue is the probability that our sample
statistic is that extreme assuming that H0
is true.
Look at Ha to calculate “What is the probability
of making 8 or fewer shots out of 20?”
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Binomcdf ( 20, .8, 8) = .000102
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Significance Test Procedure
STEP 4: Interpret the results in the
context of the problem.
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You reject H0 because the probability of being an 80%
shooter and making only 8 out of 20 shots is extremely
low. You conclude that Ha is correct; the true proportion
is < 80%
There are only two possibilities at this
step:
You reject H0 because the probability is so low. We
accept Ha
You fail to reject H0 because the probability is not low
enough
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Basic Idea Behind Significance
Tests
An outcome that would rarely
happen if a claim were true is
good evidence that the claim is
not true.
Situation
Read and summarize the
situation given in:
EXAMPLE 10.9, pg 560.
Significance Test Procedure
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Step 1: Define the population and parameter
of interest. State null and alternative
hypotheses in words and symbols.
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Population: Diet cola.
Parameter of interest: mean sweetness loss.
Suppose there is no sweetness loss (Nothing
special going on). H0: µ=0.
You are trying to find if there was sweetness loss.
Your alternate hypothesis is: Ha: µ>0.
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Significance Test Procedure
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Step 2: Choose the appropriate inference procedure.
Verify the conditions for using the selected
procedure.
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We are going to use sample mean distribution:
Do the samples come from an SRS?
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Is the population at least ten times the sample size?
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We don’t know.
Yes.
Is the population normally distributed or is the sample size
at least 30.
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We don’t know if the population is normally distributed, and the
sample is not big enough for CLT to come into play. However,
the book did say “from long experience we know that individual
tasters’ scores vary according to a normal distribution.”
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Significance Test Procedure
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Step 3: Calculate the test statistic and the Pvalue. The P-value is the probability that our
sample statistics is that extreme assuming
that H0 is true.
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µ=0, x-bar=1.02, σ=1
Look at Ha to calculate “What is the probability of
having a sample mean greater than 1.02?”
z=(1.02-0)/(1/root(10))=3.226,
P(Z>3.226) =.000619=normalcdf(3.226,1E99)
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Significance Test Procedure
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Step 4: Interpret the results in the
context of the problem.
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You reject H0 because the probability of
having a sample mean of 1.02 is very
small. We therefore accept the alternate
hypothesis; we think the colas lost
sweetness.
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REMEMBER ! ! ! ! ! !
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Hypothesis tests (or Significance
Tests) find p-values.
P-values describe how probable the
NULL HYPOTHESIS is based on the
sample statistic.
WE ARE ALWAYS TESTING
WHETHER THE NULL HYPOTHESIS
IS PROBABLE OR NOT!
Check for Understanding
A hypothesis test was calculated. Answer
whether there is significant evidence to
reject the H0 and accept the the Ha.
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P-value
P-value
P-value
P-value
P-value
=
=
=
=
=
.45
.03
.99
.10
.11
P-value = .0001
P-value = .21
P-value = 4.2341E-12
Writing Your Conclusion
There is a (p-value) probability that our
sample statistic of (sample mean or
proportion) would occur if the Null
Hypothesis were true and the population
parameter was (pop. Mean or prop).
This suggests that Null hypothesis is
(likely, somewhat likely, not likely). If
not likely, then describe the alternative
hyp. As more likely.
Statistically Significant
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To describe how significant the
evidence against the H0 is, we establish
a significance level ().
If the p-value is as small or smaller than
the established , we say the evidence
to reject the H0 is statistically
significant at the -level.
Statistical Significance
The smaller the significance level, the
more evidence against the H0 (or the
more likely the Ha).
Significance does not mean
“important”; it means that the outcome
is not likely to occur just by chance.
Check for Understanding
If the following p-values were calculated,
state whether there is statistically
significant evidence against the H0 at the
0.10, 0.05, or 0.01 level.
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P-value
P-value
P-value
P-value
P-value
=
=
=
=
=
.45
.03
.99
.10
.11
P-value = .0001
P-value = .21
P-value = 4.2341E-12
Summary
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We
We
We
We
stated a null hypothesis (no loss)
calculated a test statistic.
stated alternative hypothesis.
found the probability of getting the
test statistic if H0 was true. (p-value)
Since p-value was very low, it was
statistically significant evidence that the
null hypothesis was false and the
alternative true.