Download Inference Toolbox for Significance Tests

AP Statistics Section 11.2 A
Inference Toolbox for Significance
Tests
The four-step Inference Toolbox
will once again guide us through
the inference procedure.
Inference Toolbox for Significance Tests
Step 1: Hypothesis: Identify the population of interest and the
parameter you want to draw conclusions about. State
hypotheses.
Step 2: Conditions: Choose the appropriate inference procedure.
Verify the conditions for using it.
Step 3: Calculations: Carry out the inference procedure by
test statistic
calculating the _______________and
find the ____________.
p - value
Step 4: Interpretation: Interpret your results in the context of
the problem by interpreting the p-value or make a decision
H 0 using statistical significance. Don’t forget the three
about _____
C’s: ____________,
conclusion ____________
connection and __________.
context
z-Test for a Population Mean
To test the hypothesis based on an SRS of size n from a
population with unknown mean  and known standard deviation  ,
x  0
compute the one-sample z-statistic.
z
s
n
To determine the p-value, compute the probability of getting a value at
least as extreme as the value of our test statistic. The alternative
hypothesis ( H a) tells us if we are right-tailed, left-tailed or two-tailed.
These P-values are exact if the population is Normal and are
large n in other cases.
approximately correct for __________
Example 1: The medical director of a large company is
concerned about the effects of stress on the company’s
younger executives. According to the National Center
for Health Statistics, the mean systolic blood pressure
for males 35 to 44 years of age is 128, and the standard
deviation in this population is 15. The medical director
examines the medical records of 72 male executives in
this age group and finds that their mean systolic blood
pressure is x  129.93. Is this evidence that the mean
blood pressure for all the company’s younger male
executives is different from the national average?
(Assume that the executives have the same  as the
general population.)
Hypothesis:
The population of interest is
middle - age executives at this company.
We wish to test H 0 :   128 H a :   128
Where  is the mean systolic blood pressure for the population
Conditions:
SRS Seems safe to assume the 72 were randomly chosen. If not
an SRS, results may not generalize to the population.
Normality of x :
n  72 means the CLT will give x a distribution that is
approximately Normal
Independence:
Since we are sampling without replacement,
we must asume that N  10(72) or 720.
Calculations:
129.93 - 128
z
 1.09
15
72
p  value  2(.1379)  .2758
Interpretation:
If we assume the population mean is 128, there is a 27.58% chance of a
random sample having a mean systolic blood pressure as extreme as 129.93.
There is very little evidence that the mean systolic blood pressure of young
execs at this company differs from the national average.
Note: The data in Example 1 do not
establish that the mean blood
pressure for this company’s
middle-aged male executives is
128. We simply failed to find
convincing evidence that the mean
differed from 128.
Failing to find evidence against H0
means only that the data are
consistent with H0, not that we
have clear evidence that H0 is
true.
Hypothesis:
The population of interest is all employees at this company.
We want to test H 0 :   0 vs H a :   0
Where  is the mean change in blood pressure of employees.
Conditions:
SRS: Director chose a random sample of 50 employees.
Normality of x :
n  50 means the CLT gives a distribution for x that is approx. Normal
Independence:
Since we are sampling without replacement, we must
assume that the population of all employees  10(50) or 500.
Calculations
60
z
 2.12
20
50
p  value  .017
Interpretation:
Since our p - value of .017 is less than the significance level of .05,
we will reject the H 0 . We conclude that the mean change in blood
pressure for the employees at this company is less than 0.