Transcript AP Statistics: Section 12.2

AP Statistics: Section 12.2
We now turn out attention to conducting a significance
test about a population proportion. Recall the
conditions necessary to construct a confidence interval
for a population proportion, because the conditions for
conducting a significance test about a population
proportion are the same.
SRS
np  10 and n 1 - p   10
Normality for distribution: ____________________
Independence
Our general form for our test statistic has
always been
sample statistic - hypothesized value
Test statistic = -------------------------------------standard error
(Also called a large - sample test for a proportion)
pˆ  p0
p0 (1  p0 )
n
n(1  p0 )
z  distribution
np0
Example: An experiment on the side effects of
pain relievers randomly assigned arthritis
patients to one of several over-the-counter pain
medications. Of the 440 patients who took pain
reliever A, 23 suffered some “adverse effect”.
a) Does this experiment provide strong evidence
that fewer than 10% of the people who take this
medication have adverse effects?
Hypothesis:
The population of interest is people who take pain reliever A.
H 0 : p  .1 H a : p  .1
p  the proportion of people taking
pain reliever A who have adverse effects.
Conditions:
SRS : An experiment often uses volunteers. If the sample is not an SRS
results may not generalize to the population.
Normality of pˆ : 440(.1)  44  10 and 440(.9)  396  10
Independence : The randomization in an experiment helps with
independence. And, since sampling w/o replacement N  10n
Calculations:
z
 .1
440
 3.34
(.1)(.9)
440
23
TI83/84 : STAT TESTS 5 : 1 - PropZTest
Interpretation:
Our small p - value of .0004 indicates that there is only a .04% chance
of getting a sample with a proportion of 23
or smaller when we assume
440
the pop[ulation proportion is .1. This is strong evidence that the true population
proportion is actually less than .1.
b) Describe a Type I error and a Type II in this
situation and give consequences of each.
Type I : Assume the population proportion is less than .1 when it is not.
Consequence would be believing the drug is safer than it is
Type II : Assume the population proportion is not less than .1 when it actually is.
Consequence might be reduced sales or having the FDA pull the drug.
Note I: The standard error for the confidence interval is
computed using pˆ , while the denominator for the test
statistic is computed using the value in the null
hypothesis p0. Consequently, the correspondence
between a two-tailed significance test and a confidence
interval for a population proportion is no longer exact.
However, they are still very close.
Note II: Confidence Intervals provide
information that significant tests do not –
namely, a range of plausible values for the
true population proportion.