Transcript The z test statistic & two-sided tests

The z test statistic
&
two-sided tests
Section 10.2.3
Starter 10.2.3
• Write a definition of  as used in hypothesis
tests
Today’s Objectives
• Calculate the z test statistic for use in a hypothesis
test
• Use the z test statistic in a two-sided test
• Use a confidence interval to perform a two-sided
test
California Standards
17.0 Students determine confidence intervals for a simple
random sample from a normal distribution of data and
determine the sample size required for a desired margin of
error.
18.0 Students determine the P- value for a statistic for a
simple random sample from a normal distribution.
The z test statistic
• Recall that observations of a variable which
varies normally can be standardized into a
x
z-score by the formula z 

• If the observation used in this formula is a
sample mean (), then the result is called
the z test statistic and the formula becomes
z
x  o
/ n
P-value and the z test statistic
• Remember that P-value is the
probability of getting results as
extreme as (or more extreme
than) those actually achieved if
Ho is true
• So the P-value is simply the
area to the right (or left) of the
z test statistic under the
standard normal curve
– Area to the right if Ha is µ > µo
(where µo is the assumed mean)
– Area to the left if Ha is µ < µo
The two-sided Hypothesis Test
• When Ha is that µ is
different from some
specific value, the P-value
has to include areas both
right and left of the
observation
– That’s because the question
we ask is how likely are
results this far (or farther)
from the assumed mean, not
results this much greater
than or this much less than
the assumed mean
• So P is the total area in the
picture is 2 x .045 = .090
Example 10.11
• The mean blood pressure for American males 35
to 44 years of age is 128 with standard deviation
of 15.
• Is there evidence at the 5% significance level that
a certain company’s executives’ blood pressure
differs from the national average?
• Write hypotheses and significance level.
– Ho: µ = 128
– Ha: µ  128
–  = .05 (an arbitrary choice, but not unreasonable)
• A sample of 72 executives at the company has a
mean blood pressure of 126.07.
• Calculate the z-test statistic and P-value and write
a conclusion.
Answer
• Calculate the z test statistic.
z = (126.07 – 128) / (15 / √72) = -1.09
• Find the P-value.
P = 2 x normalcdf(-999, -1.09) = 2 x .138 = .276
• Note: 2 x normalcdf(0, 126.07, 128, 1.768) is not
recommended. The AP test readers want to see the test
statistic AND the P-value calculated.
• Write a conclusion in proper form.
Because P > , there is not sufficient evidence to
support the claim that the executives’ mean blood
pressure differs from the national average.
Confidence intervals and two-sided tests
• Suppose we believe that the true mean of a certain
population is µo.
• Someone says we are wrong – that the mean is
different from µo.
– Note that he does not have to say whether it is too great
or too small, just that it is different from µo.
• We take a sample, find , and calculate a level C
confidence interval.
• It turns out that the interval does not contain µo.
• So is he right?
– Probably yes, because if the mean really is µo then C%
of all intervals would contain µo and this one did not!
How does that compare to a
hypothesis test?
• Hypotheses & significance level
– Ho: µ = µo
Ha: µ  µo
Let  = 1 – C
• If it is true that µ = µo, then in C% of all samples,
we will fail to reject Ho.
• This is just like the statement we made about
confidence intervals:
– If the mean really is µo then C% of all intervals would
contain µo.
• So a level C confidence interval can give exactly
the same result as a two-sided hypothesis test
where  = 1 – C.
Revisit the blood pressure example
• Use the facts from the previous example and the Zinterval
screen on the TI to form a 95% confidence interval.
– n = 72
 = 15
 = 126.07
(Notice that C = 1 -  because we used α = .05)
• You should find that the C.I. is (122.61, 129.53)
• Notice that the interval contains the assumed mean of 128
• So at the 95% confidence level there is not sufficient
evidence to reject Ho
• That is the same result we got in the hypothesis test
– Caution: This approach only works for two-sided tests (where Ha is µ  µo)
because confidence intervals are essentially two-sided in that they are built
symmetrically around 
Homework
• Read pages 544 – 549
• Read pages 554 – 555
• Do problems 39, 41 – 44