Transcript AP Statistics Chapter 21 Notes

AP Statistics
Chapter 21 Notes
“More about Hypothesis
Testing”
“P-Value”
• The “p-value” of a hypothesis test is the probability of
your sample’s results occurring by natural sampling
variability.
• It is NOT the probability that the null hypothesis is true.
• Example: Interpret p = 7%
– Correct: There is a 7% chance of this sample’s results
occurring naturally if the null is true.
– Incorrect: There is a 7% chance that the null hypothesis is
true.
• When the “p-value” is small (usually less than 5%), it
tells us that it is not likely that our sample’s results
occurred naturally and therefore the null hypothesis
should be rejected.
Significance Level
(A.K.A. Alpha Level)
• The significance level (or alpha level) is the threshold we
use to determine whether the results of our hypothesis
testing indicate rejecting or retaining the null hypothesis.
• 5% is the most common, however, it is possible to have
other significance levels such as 10% or 1%.
• If the “p-value” is less than the alpha level, we reject the
null hypothesis
– Say “we have sufficient evidence from our data to conclude…”
• If the “p-value” is higher than the alpha level, we retain the
null hypothesis
– Say “we have insufficient evidence from our data to
conclude…”
Types of Errors
• We are never 100% certain. There is always a potential
for error. Here are the types of errors we can make:
• Type 1 Error – You rejected the null hypothesis but it was
actually true.
• Type 2 Error – You retained the null hypothesis but it was
actually incorrect.
Example
• Production managers on an assembly line must monitor the
output to be sure that no more than 2% of their products
are defective. They periodically inspect a random sample
of the items produced. Based on the results of their
sample, they will shut down the assembly line if they
believe that more than 2% of the items produced are
defective.
• State the hypotheses.
• In this situation, what is a type 1 error? Why is this bad?
• In this situation, what is a type 2 error? Why is this bad?
Example
•
A statistics professor has observed that for several years about 13% of the
students who initially enroll in his introductory statistics course withdraw
before the end of the semester. A salesman suggests that he try a certain
software package that gets students more involved with computers,
predicting that it will lower the dropout rate.
•
1. What are the null and alternative hypotheses?
•
2. In this situation, what is a type 1 error? Why is this bad?
•
3. In this situation, what is a type 2 error? Why is this bad?
•
4. Initially, 203 students signed up for his introductory statistics course
and 11 dropped out before the end of the semester. Perform a
significance test at the 5% level. Should the professor spend money to
continue using this software?
The Power of the Test
• The power of the test is the potential for the null
hypothesis to be rejected.
• A higher power means more potential to reject the null
hypothesis
• You are not asked to calculate the value of the power of
the test in this course, just to understand the factors that
influence it.
Factors that influence the
power of the test
• Higher significance level = stronger
power because there is more chance to
reject the null
– 10% alpha level has a stronger power than
a 5% alpha level
• Larger sample size = stronger power
because it reduces the standard
deviation which decreases the
probability of a type 2 error
– A sample size of 1000 has a stronger power
than a sample size of 500
Back to the last example…
• How will the power of the test be
influenced if the professor uses a 1%
significance level rather than a 5%
significance level?
• How will the power of the test be
influenced if the professor uses a
larger sample of statistics students?