Transcript HYPOTHESIS TESTING: A FORM OF STATISTICAL INFERENCE

HYPOTHESIS TESTING: A FORM
OF STATISTICAL INFERENCE
Mrs. Watkins
AP Statistics
Chapters 23,20,21
What is a hypothesis test?
• Hypothesis Testing: Method for
using sample data to decide between
2 competing claims about a
population parameter (mean or
proportion)
What question do such tests answer?
Is our finding due to chance or is it
likely that something about the
population seems to have
changed?
Why Statistical Inference?
The only way to “prove” anything is to
use entire population, which is not
possible.
So, we use INFERENCE to make
decisions about a population, based
on a sample
EXAMPLE:
• A new cold medicine claims to reduce the
amount of time a person suffers with a cold. A
random sample of 25 people took the new
medicine when they felt the onset of a cold and
continue to take it twice a day until they
felt better. The average time these people took
the medication was 5.2 days with a standard
deviation of 1.4 days. The typical time a person
suffers with a cold is said to be one week.
Questions from our cold study:
What is the difference between the population
mean and the sample mean? 1.8 days
Is this difference likely to be due to chance?
How could we compute how likely it is
to see a mean of 5.2 when we are
expecting a mean of 7 days?
Use a z score!
z = 7 – 5.2_ = 6.43
1.4/√25
probability of this is nearly 0…so unlikely
Hypotheses:
Ho: μ = 7 (the status quo of cold duration)
Ha: μ < 7 (what we hope to be true about the
new medication)
Our evidence suggests that Ha is more likely to
be true.
Writing Hypotheses:
Statistical Hypothesis: a claim or
statement about the value of the
population parameter
2 Hypotheses:
• Null Hypothesis: claim that is assumed to be
true—usually based on past research
Noted Ho
• Alternative Hypothesis: competing
claim based on a new sample suggesting
that a change has occurred
Noted Ha
Kinds of Tests:
Two tailed: Ho: μ = 7
Ha: μ ≠ 7
Right tailed: Ho: μ = 7
Ha: μ > 7
Left tailed: Ho: μ = 7
Ha: μ < 7
Hypotheses Example 1:
A medical researcher wants to know if
a new medicine will have an effect
on a patient’s pulse rate. He knows
that the mean pulse rate for this
population is 82 beats per minute:
Ho: μ = 82
Ha: μ ≠ 82
Hypotheses Example 2:
A chemist invents an additive to
increase the life of an automobile
battery. The mean lifetimes of a
typical car battery is 36 months.
Ho:μ = 36
Ha:μ > 36
Hypotheses Example 3:
An educational research group is
investigating the effects of poverty on
elementary school reading levels. Prior
research suggests that only 46% of
children from poor families achieve grade
level reading by third grade
Ho: p = 0.46
Ha: p ≠ 0.46
Hypotheses Example 4:
A cancer research team has been given
the task of evaluating a new laser
treatment for tumors. The current
standard treatment is costly and has
a success rate of 0.30.
Ho: p = 0.30
Ha: p > 0.30
Statistical Significance:
The results of an experiment or
observational study are too
“different” from the established
population parameter to have
occurred simply due to chance….
Something else must be going on…..
ASSIGNMENT:
Now go on-line and watch this video carefully
for good example of hypothesis testing in use:
http://www.learner.org/courses/againstallodds/
unitpages/unit25.html
α = rejection region
α is the rejection region on the normal curve,
accepted to be the highest probability that
cause you to uphold the Ho.
RESULTS OF HYPOTHESES TESTS
Let’s assume α = 0.05.
If p < α, then we reject Ho.
The sample result is too
unlikely to have happened due to
chance, so the Ho is overturned.
If p > α, then we fail to reject Ho.
The sample result could have
happened due to chance, so the
Ho is upheld.
What does p value mean?
The p value is the probability (based on z
or t curve) of seeing a sample mean of
this value or more extreme if the Ho is
really true.
If p value is low, then the Ho must not be
true. The sample data suggests that the
status quo has changed.
Conclusions of Hypothesis Tests
Rejecting Ho =
Statistically significant change
Failing to reject Ho=
Difference between sample mean
and Ho mean was not statistically
significant.
Testing about Means
When investigating whether a claim about a
MEAN is correct, you have to decide whether to
do a t test or a z test.
Z test: if you know pop. standard deviation
T test: if you know sample standard deviation
HYPOTHESIS TESTS
H: Hypotheses
A: Assumptions
T: Test and Test Statistic
P: P value
I: Interpretation of p value
C: Conclusion
HYPOTHESIS TESTING FOR
PROPORTIONS
EXAMPLE
A newspaper article from 5 years ago claimed
that 9.5% of college students seriously
considered suicide sometime during the
previous year. If a sample from this year
consisted of 1,000 students and 144 claimed
that they had seriously considered suicide, is
there evidence to suggest that the proportion
has increased?
DRAW THE MODEL OF THE SAMPLING
DISTRIBUTION OF THE PROPORTION
Hypotheses
Null Hypothesis: Ho : p = 0.095
(the stated claim about the population
proportion)
Alternative Hypothesis:
Ha: p > 0.095
Ha: p < 0.095
Ha: p ≠ 0.095
Z Proportion Test
There are no t tests for proportions, only z.
Note that we are using p not 𝒑 for
standard deviation of distribution.
Test Statistic: z
=
𝑝 −𝑝
𝑝𝑞
𝑛
P value: use normal cdf
Assumptions:
1. Random Sample or Random Assignment
2. Large enough to assume normal model:
n p > 10
n q > 10
Note that we are using p not 𝒑 for
verifying normality assumption.
EXAMPLE: DO HATPIC
An educator claims the dropout rate in
Ohio schools is 15%. Last year, 280
seniors from a random sample of
2000 seniors withdrew from school.
At α = 0.05, can the claim of 15% be
supported or is the proportion
statistically significantly different?