Transcript P - Bibb County Schools

SECTION 12.1
Tests About a Population Mean
What’s the difference between
what is addressed in Section
11.2 (we skipped) and what we
are beginning in Section 12.1?
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In reality, the standard deviation σ of the
population is unknown, so the procedures
from last chapter are not useful. However,
the understanding of the logic of the
procedures will continue to be of use.
order to be more realistic, σ is estimated from
the data collected using s
 In
Overview of a Significance Test
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A test of significance is intended to assess the evidence
provided by data against a null hypothesis H0 in favor of
an alternate hypothesis Ha.
The statement being tested in a test of significance is
called the null hypothesis. Usually the null hypothesis is
a statement of “no effect” or “no difference.”
A one-sided alternate hypothesis exists when we are
interested only in deviations from the null hypothesis in
one direction
H0 : =0
Ha : >0 (or <0)
If the problem does not specify the direction of the
difference, the alternate hypothesis is two-sided
H0: =0
Ha: ≠0
HYPOTHESES
 NOTE:
Hypotheses ALWAYS
refer to a population parameter,
not a sample statistic.
 The alternative hypothesis should
express the hopes or suspicions
we have BEFORE we see the
data. Don’t “cheat” by looking at
the data first.
CONDITIONS
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These should be VERY FAMILIAR to you by now.
 Random
 Data is from an SRS or from a randomized experiment
 Normal
 For means—population distribution is Normal or you have
a large sample size (n≥30) to ensure a Normal sampling
distribution for the sample mean
 For proportions—np≥10 and n(1-p)≥10 (meaning the
sample is large enough to ensure a Normal sampling
distribution for p̂ )—more details in next Section 12.2
 Independent
 Either you are sampling with replacement or you have a
population at least 10 times as big as the sample to
make using the formula for st. dev. okay.
CAUTION
 Be
sure to check that the conditions
for running a significance test for
the population mean are satisfied
before you perform any
calculations.
t Statistic
note: this is the same as what we learned in Chapter 10
x  0
t
s
n
The statistic does not have a normal distribution
 Degrees of freedom: n-1
 Differs from a z statistic because σ is not used
 t statistic says how far x is from its mean μ in
standard deviation units
 We are now using the body of the table we used in
the last chapter.
ROBUSTNESS
ROBUST: Confidence levels or P-Values
do not change when some of the
assumptions are violated
 Fortunately for us, the t-procedures are
robust in certain situations.
 Therefore . . .
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This is when we use the t-procedures
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It’s more important for the data to be
an SRS from a population than the population
has a normal distribution
If n is less than 15, the data must be normal to
use t-procedures
If n is at least 15, the t-procedures can be used
except if there are outliers or strong skewness
If n≥40, t-procedures can be used even in the
presence of strong skewness
Density Curves for
t Distributions
Bell-shaped and symmetric
 Greater spread than a normal curve
 As degrees of freedom (or sample size)
increases, the t density curves appear
more like a normal curve
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Tip on Interpreting P-values
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On page 746, at the bottom of the box “The OneSample t Test,” it is stated, “These P-values are
exact if the population distribution is Normal and
are approximately correct for large n in other
cases.” Example 12.2 (same page) then uses a
small sample (n=10) with no guarantee that the
distribution is Normal and where the Normal
probability plot is a bit “iffy.” Thus, this is a case in
which we choose to use t where we assume that
the population distribution is approximately Normal
because we don’t have clear evidence that it isn’t.
Therefore, the P-values are approximately correct.
INFERENCE TOOLBOX (p 705)
DO YOU REMEMBER WHAT THE STEPS ARE???
Steps for completing a SIGNIFICANCE TEST:
 1—PARAMETER—Identify the population of interest
and the parameter you want to draw a conclusion
about. STATE YOUR HYPOTHESES!
 2—CONDITIONS—Choose the appropriate inference
procedure. VERIFY conditions (Random, Normal,
Independent) before using it.
 3—CALCULATIONS—If the conditions are met, carry
out the inference procedure.
 4—INTERPRETATION—Interpret your results in the
context of the problem. CONCLUSION,
CONNECTION, CONTEXT(meaning that our
conclusion about the parameter connects to our work
in part 3 and includes appropriate context)
Step 1—PARAMETER
Read through the problem and determine
what we hope to show through our test.
 Our null hypothesis is that no change has
occurred or that no difference is evident.
 Our alternative hypothesis can be either
one or two sided.
 Be certain to use appropriate symbols and
also write them out in words.
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Step 2—CONDITIONS
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Based on the given information, determine which test
should be used. Name the procedure.
State the conditions.
Verify (through discussion) whether the conditions
have been met. For any assumptions that seem
unsafe to verify as met, explain why. Don’t forget, with
the t distribution, there is more “forgiveness” due to
the robustness of the t procedures
Remember, if data is given, graph it to help facilitate
this discussion
For each procedure there are several things that we
are assuming are true that allow these procedures to
produce meaningful results.
Step 3—CALCULATIONS
First write out the formula for the test statistic,
report its value, mark the value on the curve.
 Sketch the density curve as clearly as possible
out to three standard deviations on each side.
 Mark the null hypothesis and sample statistic
clearly on the curve.
 Calculate and report the P-value
 Shade the appropriate region of the curve.
 Report other values of importance (standard
deviation, df, critical value, etc.)
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Step 4—INTERPRETATION
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There are really two parts to this step: decision &
conclusion. TWO UNIQUE SENTENCES.
Based on the P-value, make a decision. Will you
reject H0 or fail to reject H0.
If there is a predetermined significance level, then
make reference to this as part of your decision. If not,
interpret the P-value appropriately.
Now that you have made a decision, state a
conclusion IN THE CONTEXT of the problem.
This does not need to, and probably should not, have
statistical terminology involved. DO NOT use the
word “prove” in this statement.
The Steps for a
ONE SAMPLE t-TEST
Same Approach—Slightly Different Look
STATE
State the hypotheses and name test
1.
Ho:  = 0
Ha:  ‹, ›, or ≠ 0
PLAN
State and verify your assumptions
Calculate the P-value and other important values
2.
DO
3.
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Done in calculator or…
Book:
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CONCLUDE
4.
Using Table C, look in the df (n-1) column and then look
across the line to find the range of probabilities the t statistic
falls in
State Conclusions (Both statistically and contextually)
- The smaller the P-value, the greater the evidence is to
reject Ho
Summarizing the STEPS of Inference
State the null and alternative hypotheses
in context
 Identify the inference procedure to be
used and justify the conditions for its use
 Perform statistical mechanics
 State the conclusion in the context of the
problem with a clear linkage to the
mechanics that imply that conclusion
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Example 1-sided t-Test
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The diastolic blood pressure for American women
aged 18-44 has approximately the Normal distribution
with mean =75 milliliters of mercury (mL Hg) and
standard deviation s=10 mL Hg. We suspect that
regular exercise will lower blood pressure. A
random sample of 25 women who jog at least
five miles a week gives sample mean blood
pressure x =71 mL Hg. Is this good evidence
that the mean diastolic blood pressure for the
population of regular exercisers is lower than 75
mL Hg?
Step 1
The parameter of interest is the mean diastolic
blood pressure .
 Our null hypothesis is that the blood pressure is
no different for those that exercise.
 Our alternative hypothesis is one-sided
because we suspect that exercisers have lower
blood pressure.
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H0:  = 75 mL
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Ha:  < 75 mL
Step 2
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Since we do not know the population standard
deviation we will be performing a t-test of significance.
We were told that the sample is random, but we do not
know if it is an SRS from the population of interest.
This may limit our ability to generalize.
Since the population distribution is approximately
Normal, we know that the sampling distribution of x
will also be approximately Normal. So we are safe
using the t procedures.
The blood pressure measurements for the 25 joggers
should be independent. Note that the population of
interest is at least 10 times as large as the sample.
Step 3
A curve should be drawn, labeled, and shaded.
 You can use the formula to calculate your t test
statistic for this problem
x
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
0
 t
In this case t = -2.00
s
n
 Mark this on your sketch.
 Based on our calculations the P-value is
0.0285.
x  71 , s=10, n=25
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Step 4
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Since there is no predetermined level of significance
if we are seeking to make a decision, this could be
argued either way. If exercisers are no different, we
would get results this small or smaller about 2.85% of
the time by chance.
This result is significant at the 5% level, but is not
signficant at the 1% level.
We would likely reject H0.
There is not much chance of obtaining a sample like
we did if there is no difference, so we would reject the
idea that there is no difference and conclude that the
mean diastolic blood pressure of American women
aged 18-44 that exercise regularly is probably less
than 75 mL Hg.
DUALITY
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A level α two-sided significance test rejects a
hypothesis H0 : = 0 exactly when 0 falls outside
a level 1- α confidence interval for .
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This relationship is EXACT for a TWO-SIDED
hypothesis test FOR A MEAN, but IS NOT EXACT
FOR tests involving PROPORTIONS.
Essentially, if the parameter value given in the null
hypothesis falls inside the confidence interval, then
that value is plausible. If the parameter value lands
outside the confidence interval, then we have good
reason to doubt H0.
Matched Pairs
(Paired t Tests)
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To compare the responses to the two
treatments in a matched pairs design,
apply the one sample t procedures to the
observed differences
More commonly used than single-sample
studies
 Use calculator
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Example: Lean vs. Obese
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Some studies have shown that lean and obese people spend
their time differently. Obese people spend fewer minutes per
day (on average) standing and walking than do lean people
who are similar in age, overall health, and occupation. Is this
difference biological, so that it might help explain why some
people become obese? Or is it a response to obesity—people
become less active when they gain weight?
A small pilot study looked at this issue. The subjects were 7
mildly obese people who were healthy and did not follow an
exercise program. The subjects agreed to participate in a
weight-loss program for eight weeks, during which they lost and
average of 8 kilograms (17.6 pounds). Both before and after
weight loss, each subject wore monitors that recorded every
movement for 10 days. The table on the next slide shows the
minutes per day spent standing and walking. The response
variable for this study is the difference in minutes after weight
loss minus minutes before weight loss. The differences appear
in the final column of the table.
Time standing and walking before
and after weight loss
Subject
1
2
3
4
5
6
7
Minutes per Day
Before
After
Difference
293
264
-29
330
335
5
353
387
34
354
307
-47
400
387
-13
454
358
-96
552
549
-3
Step 1—Parameter
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Researcher’s question: Do mildly obese people
increase the time they spend standing and walking
when they lose weight?
The parameter of interest is the mean difference
(after-before)  in activity time in the entire
population of such mildly obese people. The null
hypothesis is “no change.” That is, the mean
difference in the entire population of mildly obese
people who lose weight is zero. The alternative
hypothesis is that these people will increase their
activity after weight loss and therefore have a
positive difference.
Ho:  = 0
Ha:  > 0
Step 2—Conditions
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Random—The 7 subjects volunteered. We
must be willing to assume that they are a
random sample from all people who meet
requirements for the study (mildly obese,
healthy, sedentary jobs, no exercise program,
etc.). Human subjects are almost never
actually chosen at random from the population
of interest, so this study is typical. We rely on
researchers not to bias their study by their way
of choosing subjects.
Step 2 Continued—Conditions
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Normality—The difference -96 for Subject 6 may be a
low outlier (although it passes our standard 1.5*IQR
rule). Because the observations are widely spread, it
is hard to judge normality from just 7 observations.
The Normal probability plot suggests that these data
could come from a Normal population.
Independence—The differences in standing and
walking time for these 7 subjects should be
independent. Also, there are probably at least 70
people that fall into this population allowing us to
assume independence.
NOTE: The before and after measurements for each
subject are NOT independent, which is why we use a
paired T-test.
Step 3—Calculations
We are performing a matched pairs t-Test.
x  0
 t = -1.3492
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t
s
n
df = n-1 = 7-1 = 6
 P-value = 0.8870
 x = -21.2857
 s = 41.7401
n=7
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Don’t forget to draw your
curve. Remember, this is no
longer a Normal curve.
Instead, we have a curve for
the t-distribution. For
drawing this, look at your
calculator and remember it is
nearly the Normal curve.
Step 4—Interpretation
With a P-value this high, we would fail to reject
H0 at any reasonable significance level. The
mean difference in activity time in the
population of mildly obese people could very
well be 0. It seems that having mildly obese
people lose weight may not increase their
activity time.
 NOTE: This is an unusual case where the
value from our sample is in the opposite
direction from our alternative hypothesis.
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COMPUTER OUTPUT
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Unfortunately, we rarely (or never) get the chance to
use a computer to analyze data. However, you are
expected to be able to read computer output for the
purposes of this course as well as for the AP Exam in
May.
The book provides several examples for you to use in
your efforts to understand computer output.
We will occasionally see additional examples in class.
Most computer output is similar, so make sure you
know what you are looking for.
Most computer output also has many numbers that
you will not use, so make sure you know which
numbers matter and which ones do not.
Another Example
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A medical researcher wishes to investigate the
effectiveness of exercise versus diet in losing weight.
Two groups of 25 overweight adult subjects are used,
with a subject in each group matched to a similar
subject in the other group on the basis of a number of
physiological variables. One of the groups is placed
on a regular program of vigorous exercise but with no
restriction on diet, and the other on a strict diet but
with no requirement to exercise. The weight losses
after 20 weeks are determined for each subject, and
the differences between matched pairs of subjects
(weight loss of subject in exercise group – weight loss
of matched subject in diet group) is computed. The
mean of these differences in weight loss is found to be
-2 lb. with standard deviation s = 4 lb.
Is this evidence of a significant difference in mean
weight loss for the two methods?
Step 1—Parameter
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Let  be the mean difference in weight loss
(exercise – diet) where the difference is for
each pair of subjects.
H0: =0
Ha: ≠0
The null hypothesis is that there is no difference
in weight loss between the two methods.
 The alternative hypothesis is that there is a
difference in weight loss between the two
methods.
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Step 2—Conditions
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Random—We are not told how the subjects were chosen,
so we must assume they are representative of the desired
population if we want to extend our findings to that larger
population.
Normality—Recall, due to working with the t-distribution,
when the sample size is sufficiently large, we become
unconcerned with the Normality of the population
distribution. In this case, the sample size is large enough
to overcome some skewness, but we would be more
comfortable if we could safely assume Normality in the
population distribution. Of course, outliers would damage
our results.
Independence—We must be willing to view these 25
differences as independent measurements or assume that
there are at least 250 differences in the population.
Step 3—Calculations
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t = -2.5
x  0
t
s
n
df = 24
x = -2
s=4
n = 25
P-value = 0.0197
Don’t forget to draw your
curve. Remember, this is no
longer a Normal curve.
Instead, we have a curve for
the t-distribution. For
drawing this, look at your
calculator and remember it is
nearly the Normal curve.
Step 4—Interpretation
Assuming all conditions are satisfied:
 Because the P-value is small, we can reject H0.
 Essentially, if there is truly no difference
between the two methods, we would only get
differences in weight loss this extreme about
1.97% of the time by chance alone. Since this
is so unlikely, we can reject the null hypothesis.
 Based on this evidence, there appears to be a
difference in the average weight loss between
the two methods.
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Follow Up Question to Example
Can we conclude that a significant difference
in weight loss for the two methods is CAUSED
by the specific treatment administered (diet or
exercise)? Justify your answer.
 Assuming the subjects are randomly assigned
to each of the weight loss groups, then cause
and effect conclusions can be drawn from this
matched pairs experiment. For example, once
the pairings are made, the toss of a coin (or
other random event) should determine which
subject of each pair goes on which program.
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