Download 8.2 Testing a Proportion

Section 8.2 Testing a Proportion DISCUSSION
Statistics
A Test of _________________ : comparing results from a sample to some
_________________ .
For example, say that a person in a town claims that 70% of residents in the
town support the building of a stadium, but a person conducting a simple
random sample finds that 35% of the people support the stadium. In this case,
person’s claim of 70% would be the standard that the survey results are
compared to.
If the difference between the _________________ and the _________________
is small, we cannot ________________ the standard. However, if the
difference is large, we would have reason to believe that the
___________________ no longer holds. If there is a large difference, we say
that the difference between the sample and standard is ___________________
______________________.
We have a 4-sided die that we roll. Our claim is that the die is “fair”, meaning
that there is an equal probability of getting a 1, 2, 3, or 4. Say that we roll the
die 40 times. We would expect to get a 1 _______ times, because the
probability of getting a “1” is ________. Say that instead of getting a “1”
_____ times, we get a “1” 12 times. A _____________________ in
____________________ means that we will not expect to get exactly the same
value of _______ every time we take a sample. What we do want to determine
is what are reasonably likely values.
A sample proportion, _______, is said to be _______________
_______________ if it is not a ___________________ likely outcome when the
proposed standard is true. A statement that the proposed standard is true is
called the _________ ___________________, represented by the symbol
________. The probability associated with the null hypothesis is represented as
_________. If we suspect that our null hypothesis is true, we would expect that
our sample proportion, ________, would be in the middle 95% of all possible
values of ______ if the mean is _________.
The Test Statistic for Testing a Proportion
To determine if ______ is reasonably likely or a __________ event for a given
standard, _______, we need to check the value of the __________
____________.
This tells how many standard ___________________ the
sample proportion ________ lies from the standard, _______
Instead of only indicating if a result is ________________ significant or not, we
report a _________________.
A _____________________ for a test is the probability of seeing a result from a
random sample that is as extreme as or more extreme than the result you got
from your random sample if the _________ _________________ is
__________.
If the test statistic is statistically ________________, we would _____________
the null hypothesis.
Let us look at our claim that a four-sided die is “fair”.
Our Null Hypothesis, ______ is that the probability of getting a “1” is _______.
If we roll the die 40 times and get a “1” 12 times, should we reject the null?
If we roll the die 40 times and get a “1” 18 times, should we reject the null?
Some terminology: Critical Values and Level of Significance
We ______________ the hypothesized standard, _______, when our sample
proportion, ________, would be a ___________ event if the _______________
was true. If we are dealing with 95%, rejection of the standard would occur if
the _______ is in the outer _______% (______% in each tail). For 95%, the
boundary between rejecting and not rejecting is at z scores of ________ and
_________. The boundary values are called ______________ values
(symbolized as ________). The corresponding proportion outside of the middle
% value (_______ for the case of 95%) is called the ____________ of
__________________ (symbolized as _______). Other levels of significance
can be used.
Example (from book):
Suppose you want to reject the null hypothesis, H0, when the test statistic, z, is
in the outer 10% of the standard normal distribution – that is, with a level of
significance equal to 0.10. What should you use as critical values? Should you
reject the null hypothesis, H0, if z turns out to be 1.87?
Using Critical Values and the Level of Significance
If the value of the test statistic ____ is more extreme than the ___________
values, ________, you have chosen (or, equivalently, the ______value is less
than ______), you have evidence against the ________ Hypothesis, ______.
_____________ the null hypothesis and say that the result is
______________________ significant.
However, if the test statistic, _____, is less extreme than the ____________
value (_______) (or, equivalently, the _____-value is greater than ______), you
do not have evidence against the null hypothesis and so you cannot reject it. If
a level of significance is not specified, usually assume that ______ = 0.05 and
_______ = _________.
Components of a Significance Test for a Proportion
1.) Give the name of the test and check the conditions for its use. Three
conditions that need to be met:
 The sample is a simple random sample from a ________________
population.
 Both ______ and _______________ are at least ________.
 The population size is at least ______ times the sample size.
2.) State the _________________, defining any symbols. When testing a
proportion, the null hypothesis is
______: the percentage of successes, _____, in the population from which
the sample came is equal to ______.
The alternative hypothesis, _________, can be of 3 forms:
______: the percentage of successes, _____, in the population from which
the sample came is not _________ to _______.
______: the percentage of successes, _____, in the population from which
the sample came is _______________ than _______.
______: the percentage of successes, _____, in the population from which
the sample came is _______________ than _______.
3.) Compute the test statistic, ______, and find the critical values, ________,
and the _______________. Include a sketch that illustrates the situation.
4.) Write a conclusion. There are two parts to stating a conclusion:
 Compare the value of _____ to the predetermined critical values, or
compare the ________________ to ______. then say whether you
____________ the null hypothesis, linking your reason to the Pvalue or to the critical values.
 Tell what your __________________ means in the context of the
situation.
Example: Suppose 22 students out of a random sample of 40 students carry a
backpack to a school with 2000 students. Follow the steps below to
test the claim that 60% of the students in the school carry backpacks
to class.
a.) Check conditions for the test.
b.) Write your hypothesis
c.) Compute the test statistic, z, and find the critical values, z*, and
the P-value.
d.) Write a conclusion.
Types of Errors
There is an analogy with a jury trial in considering possible errors in
significance tests:
Defendant is Actually
Innocent
Guilty
Not Guilty
___________
___________
Guilty
___________
___________
Jury’s Decision
Null Hypothesis is Actually
True
False
Don’t Reject H0
___________
___________
Reject H0
___________
___________
Your Decision
The probability of making a type I error is equal to ___________.
Power of a Test
The ___________ of a test is the probability of rejecting the null hypothesis.
Given that we have a condition where the null hypothesis is false,
_____________ = 1 – __________________________.
If the probability of a Type II error is small, the power of the test is _________.
Three properties of power:
 Power increases as the sample size increases, all else being held constant.
 Power decreases as the value of α decreases, all else being held constant.
 Power increases when the true population proportion, p, is farther from
the hypothesized value, p0. (p-hat tends to be farther from p0, so the test
statistic tends to be larger.
Summary of Types of Error and Power
Type I Error:
Occurs when the null hypothesis is true and you reject it.
Probability of making a Type I error is equal to the significance
level, α, of the test.
To decrease the probability of a Type I error, make ______
smaller.
Changing the sample size has no effect on a Type I error.
If the null hypothesis is ____________, you cannot make a Type I
error.
Type II Error:
Occurs when the null hypothesis is false and you fail to reject it.
To decrease the probability of making a Type II error, taker a
larger sample or make the significance level, α, larger.
If the null hypothesis is true, you cannot make a Type II error.
Power
Power is the probability of rejecting the null hypothesis in cases
where the null hypothesis is false.
When the null hypothesis is false, you want to reject it and
therefore you want the power to be large. To increase
power, you can either take a larger sample or make α larger.
Different forms of the alternative hypothesis:
Ha: The percentage of successes p in the
population from which the sample
came is not p0
Ha: The percentage of successes p in the
population from which the sample
came is greater than p0
Ha: The percentage of successes p in the
population from which the sample
came is less than p0