Transcript Reasoning of Hypothesis testing

7.5: HYPOTHESES TESTS FOR
PROPORTIONS
Objective: To test claims about inferences for
proportions, under specific conditions
HYPOTHESES
• The hypotheses for proportions are similar to those for means.
• In fact, they are the same. We use p instead of µ
H0: p= 0.20
REASONING OF HYPOTHESIS
TESTING
There are four basic parts to a hypothesis test:
1. Hypotheses
2. Model
3. Mechanics
4. Conclusion
Let’s look at these parts in detail…
REASONING OF HYPOTHESIS
TESTING (CONT.)
1.
Hypotheses
•
The null hypothesis: To perform a hypothesis test, we must
first translate our question of interest into a statement about
model parameters.
•
In general, we have
H0: parameter = hypothesized value.
•
The alternative hypothesis: The alternative hypothesis, HA,
contains the values of the parameter we consider plausible
when we reject the null.
REASONING OF HYPOTHESIS
TESTING (CONT.)
1. Hypotheses Example:
A 1996 report from the U.S. Consumer Product Safety Commission claimed that at
least 90% of all American homes have at least one smoke detector. A city’s fire
department has been running a public safety campaign about smoke detectors
consisting of posters, billboards, and ads on radio and TV and in the newspaper.
The city wonders if this concerted effort has raised the local level above the 90%
national rate. Building inspectors visit 400 randomly selected homes and find that
376 have smoke detectors. Is this strong evidence that the local rate is higher than
the national rate? Set up the hypotheses.
2.
Model
REASONING OF HYPOTHESIS
TESTING (CONT.)
•
To plan a statistical hypothesis test, specify the model you will use to test the null
hypothesis and the parameter of interest.
•
All models require assumptions, so state the assumptions and check any
corresponding conditions.
•
Your conditions should conclude with a statement such as:
•
Because the conditions are satisfied, I can model the sampling distribution of the proportion
with a Normal model.
•
Watch out, though. It might be the case that your model step ends with
“Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case,
stop and reconsider (proceed with caution)
REASONING OF HYPOTHESIS
TESTING (CONT.)
2.
Model
•
Don’t forget to name your test!
•
The test about proportions is called a one-proportion z-test.
REASONING OF HYPOTHESIS
TESTING (CONT.)
2. Model (cont.)
• The conditions for the one-proportion z-test are the same as for the
one proportion z-interval. We test the hypothesis H0: p = p0
•
using the statistic
•
where
SD  p̂  
p̂  p0 

z
SD  p̂ 
p0 q0
n
• When the conditions are met and the null hypothesis is true, this
statistic follows the standard Normal model, so we can use that
model to obtain a P-value.
REASONING OF HYPOTHESIS
TESTING (CONT.)
2.
Model (cont.)
Finding the P-Value
Either use the table provided, or you may use your calculator:
• normalcdf( Lower z, upper z)
• OR use the test feature in the calculator
REASONING OF HYPOTHESIS
TESTING (CONT.)
3. Mechanics
•
Under “mechanics” we place the actual calculation of our test statistic
from the data.
•
Different tests will have different formulas and different test statistics.
•
Usually, the mechanics are handled by a statistics program or calculator,
but it’s good to know the formulas.
REASONING OF HYPOTHESIS
TESTING (CONT.)
3. Mechanics (continued)
•
The ultimate goal of the calculation is to obtain a P-value.
•
The P-value is the probability that the observed statistic value (or
an even more extreme value) could occur if the null model were
correct.
•
If the P-value is small enough, we’ll reject the null hypothesis.
•
Note: The P-value is a conditional probability—it’s the probability
that the observed results could have happened if the null hypothesis
is true.
REASONING OF HYPOTHESIS
TESTING (CONT.)
4. Conclusion
•
The conclusion in a hypothesis test is always a statement about
the null hypothesis.
•
The conclusion must state either that we reject or that we fail to
reject the null hypothesis.
•
And, as always, the conclusion should be stated in context.
STEPS FOR HYPOTHESIS TESTING
1. Check Conditions and show that you have checked these!
• Random Sample: Can we assume this?
• 10% Condition: Do you believe that your sample size is less than
10% of the population size?
• Success/Failure:
𝒏𝒑𝟎 ≥ 𝟏𝟎
and
𝒏𝒒𝟎 ≥ 𝟏𝟎
2. State the test you are about to conduct
Ex) One-proportion z-test
3. Set up your hypotheses
H 0:
H :
STEPS FOR HYPOTHESIS TESTING
(CONT.)
4. Calculate your test statistic
𝒛=
𝒑−𝒑𝟎
𝒑 𝟎 𝒒𝟎
𝒏
5. Draw a picture of your desired area under the t-model, and
calculate your P-value.
6. Make your conclusion.
When your P-value is small enough (or below α, if given), reject the
null hypothesis.
When your P-value is not small enough, fail to reject the null
hypothesis.
CALCULATOR TIPS
One Proportion Z-Test:
• Stat  TESTS
• 5: 1-Prop ZTest
• Po = hypothesized proportion
• x = number of successes
• n = sample size
• Determine the tail
• Calculate
EXAMPLE 1
A large city’s DMV claimed that 80% of candidates pass driving tests, but a survey
of 90 randomly selected local teens who had taken the test found only 61 who
passed. Does this finding suggest that the passing rate for teenagers is lower than
the DMV reported?
EXAMPLE 2 (PARTNERS)
Home field advantage –teams tend to win more often when the play at home.
Or do they? If there were no home field advantage, the home teams would win
about half of all games played. In the 2007 Major League Baseball season, there
were 2431 regular-season games. It turns out that the home team won 1319 of the
2431 games, or 54.26% of the time. Could this deviation from 50% be explained
from natural sampling variability, or is it evidence to suggest that there really is a
home field advantage, at least in professional baseball?
ASSIGNMENTS
• Day 1: 7.5-Set A Book Page # 1 – 3, 9, 11
• Day 2: 7.5-Set A Book Page # 4, 5, 12, 18
• Day 3: 7.5-Set A Book Page # 16, 22
7.5-Set B Book Page (Same as 6.5 B Book Page) # 6, 15, 17, 19