Download 7.1 * 7.4: Inference tests for means

7.1 – 7.4: HYPOTHESES TESTS FOR
MEANS
Objective: To test claims about inferences for
means, under specific conditions
HYPOTHESES
• Hypotheses are working models that we adopt temporarily.
• Our starting hypothesis is called the null hypothesis.
• The null hypothesis, that we denote by H0, specifies a population
model parameter of interest and proposes a value for that parameter.
• We usually write down the null hypothesis in the form
H0: parameter = hypothesized value.
• The alternative hypothesis, which we denote by HA, contains the values
of the parameter that we consider plausible if we reject the null
hypothesis.
HYPOTHESES (CONT.)
• The null hypothesis, specifies a population model parameter of
interest and proposes a value for that parameter.
• We might have, for example, H0: µ= 20.
• We want to compare our data to what we would expect given that H 0
is true.
• We can do this by finding out how many standard deviations away from
the proposed value we are.
• We then ask how likely it is to get results like we did if the null
hypothesis were true.
CONSIDER TRIALS
• Think about the logic of jury trials:
• To prove someone is guilty, we start by assuming they are
innocent.
• We retain that hypothesis (null hypothesis) until the facts make it
unlikely beyond a reasonable doubt.
• Then, and only then, we reject the (null) hypothesis of innocence
and declare the person guilty.
CONSIDER TRIALS (CONT.)
• The same logic used in jury trials is used in statistical tests of
hypotheses:
• We begin by assuming that a hypothesis is true.
• Next we consider whether the data are consistent with the
hypothesis.
• If they are, all we can do is retain the hypothesis we started with.
If they are not, then like a jury, we ask whether they are unlikely
beyond a reasonable doubt.
ALTERNATIVE HYPOTHESES
There are three possible alternative hypotheses:
HA: parameter < hypothesized value
HA: parameter ≠ hypothesized value
HA: parameter > hypothesized value
ALTERNATIVE HYPOTHESES
(CONT.)
•
HA: parameter ≠ value is known as a two-sided alternative
because we are equally interested in deviations on either side of
the null hypothesis value.
•
For two-sided alternatives, the P-value is the probability of
deviating in either direction from the null hypothesis value.
ALTERNATIVE HYPOTHESES
(CONT.)
• The other two alternative hypotheses are called one-sided alternatives.
• A one-sided alternative focuses on deviations from the null hypothesis
value in only one direction.
• Thus, the P-value for one-sided alternatives is the probability of
deviating only in the direction of the alternative away from the null hypothesis
value.
P-VALUES
• The statistical twist is that we can quantify our level of doubt.
• We can use the model proposed by our hypothesis to
calculate the probability that the event we have witnessed
could happen.
• That is just the probability we’re looking for—it quantifies
exactly how surprised we are to see our results.
• This probability is called a P-value.
P-VALUES (CONT.)
• When the data are consistent with the model from the null hypothesis,
the P-value is high and we are unable to reject the null hypothesis.
• In that case, we have to “retain” the null hypothesis we started with.
• We can’t claim to have proved it; instead we “fail to reject the null
hypothesis” when the data are consistent with the null hypothesis
model and in line with what we would expect from natural sampling
variability.
• If the P-value is low enough, we’ll “ reject the null hypothesis ,” since
what we observed would be very unlikely were the null model true.
P-VALUES: RETURN TO TRIALS
• If the evidence is not strong enough to reject the presumption of
innocent, the jury returns with a verdict of “not guilty.”
• The jury does not say that the defendant is innocent.
• All it says is that there is not enough evidence to convict, to
reject innocence.
• The defendant may, in fact, be innocent, but the jury has no way
to be sure.
P-VALUES: RETURN TO TRIALS
(CONT.)
• Said statistically, we will fail to reject the null hypothesis.
• We never declare the null hypothesis to be true, because we
simply do not know whether it’s true or not.
• Sometimes in this case we say that the null hypothesis has
been retained.
EXAMPLES
1.
A research team wants to know if aspirin helps to thin blood. The null hypothesis
says that it doesn’t. They test 12 patients, observe their mean blood levels, given
baseline data, and get a P-value of 0.32. They proclaim that aspirin doesn’t work.
What would you say?
2.
A weight loss drug has been tested and found the mean weight loss of patients in
a large clinical trial. Now the scientists want to see if the new, improved version
works even better. What would the null hypothesis be?
3.
The new drug is tested and the P-value is 0.0001. What would you conclude
about the new drug?
THINKING ABOUT P-VALUES
• A P-value is a conditional probability—the probability of the observed statistic
given that the null hypothesis is true.
• The P-value is NOT the probability that the null hypothesis is true.
• It’s not even the conditional probability that null hypothesis is true given the data.
• Be careful to interpret the P-value correctly.
THINKING ABOUT P-VALUES
(CONT.)
• When we see a small P-value, we could continue to believe the null
hypothesis and conclude that we just witnessed a rare event. But
instead, we trust the data and use it as evidence to reject the null
hypothesis.
• However big P-values just mean what we observed isn’t surprising.
That is, the results are now in line with our assumption that the
null hypothesis models the world, so we have no reason to reject it.
ALPHA LEVELS
• Sometimes we need to make a firm decision about whether or
not to reject the null hypothesis.
• When the P-value is small, it tells us that our data are rare
given the null hypothesis.
• How rare is “rare”?
ALPHA LEVELS (CONT.)
• We can define “rare event” arbitrarily by setting a threshold for
our P-value.
• If our P-value falls below that point, we’ll reject H0. We call such
results statistically significant.
• The threshold is called an alpha level, denoted by .
ALPHA LEVELS (CONT.)
• Common alpha levels are 0.10, 0.05, and 0.01.
• You have the option—almost the obligation—to consider your
alpha level carefully and choose an appropriate one for the
situation.
• The alpha level is also called the significance level.
• When we reject the null hypothesis, we say that the test is
“significant at that level.”
ALPHA LEVELS (CONT.)
• What can you say if the P-value does not fall below ?
• You should say that “The data have failed to provide sufficient
evidence to reject the null hypothesis.”
• Don’t say that you “accept the null hypothesis.”
• Recall that, in a jury trial, if we do not find the defendant guilty,
we say the defendant is “not guilty”—we don’t say that the
defendant is “innocent.”
ALPHA LEVELS (CONT.)
• The P-value gives the reader far more information than just stating
that you reject or fail to reject the null.
• In fact, by providing a P-value to the reader, you allow that person
to make his or her own decisions about the test.
• What you consider to be statistically significant might not be the
same as what someone else considers statistically significant.
• There is more than one alpha level that can be used, but each
test will give only one P-value.
STATISTICALLY SIGNIFICANT
• What do we mean when we say that a test is statistically
significant?
• All we mean is that the test statistic had a
P-value lower than our alpha level.
• Don’t be lulled into thinking that statistical significance carries
with it any sense of practical importance or impact.
DAY 2
REASONING OF HYPOTHESIS
TESTING
There are four basic parts to a hypothesis test:
1. Hypotheses
2. Model
3. Mechanics
4. Conclusion
Let’s look at these parts in detail…
REASONING OF HYPOTHESIS
TESTING (CONT.)
1.
Hypotheses
•
The null hypothesis: To perform a hypothesis test, we must
first translate our question of interest into a statement about
model parameters.
•
In general, we have
H0: parameter = hypothesized value.
•
The alternative hypothesis: The alternative hypothesis, HA,
contains the values of the parameter we consider plausible
when we reject the null.
2.
Model
REASONING OF HYPOTHESIS
TESTING (CONT.)
•
To plan a statistical hypothesis test, specify the model you will use to test the null
hypothesis and the parameter of interest.
•
All models require assumptions, so state the assumptions and check any
corresponding conditions.
•
Your conditions should conclude with a statement such as:
•
Because the conditions are satisfied, I can model the sampling distribution of the proportion
with a Normal model.
•
Watch out, though. It might be the case that your model step ends with
“Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case,
stop and reconsider (proceed with caution)
REASONING OF HYPOTHESIS
TESTING (CONT.)
2.
Model
•
Don’t forget to name your test!
•
The test about means is called a one-sample t-test.
REASONING OF HYPOTHESIS
TESTING (CONT.)
2.
Model (cont.)
One-sample t-test for the mean
• The conditions for the one-sample t-test for the mean are the same as for
the one-sample t-interval.
• We test the hypothesis H0:  = 0 using the statistic
𝒕𝒏−𝟏 =
𝒙−𝝁𝟎
𝑺𝑬(𝒙)
• The standard error of the sample mean is 𝑺𝑬
𝒙 =
𝒔
𝒏
• When the conditions are met and the null hypothesis is true, this statistic
follows a Student’s t model with n – 1 df. We use that model to obtain a Pvalue.
REASONING OF HYPOTHESIS
TESTING (CONT.)
2.
Model (cont.)
Finding the P-Value
Either use the table provided, or you may use your calculator:
• normalcdf( is used for z-scores (if you know 𝜎)
• tcdf( is used for critical t-values (when you use s to estimate 𝜎)
• 2nd  Distribution
• tcdf(lower bound, upper bound, degrees of freedom)
REASONING OF HYPOTHESIS
TESTING (CONT.)
3. Mechanics
•
Under “mechanics” we place the actual calculation of our test statistic
from the data.
•
Different tests will have different formulas and different test statistics.
•
Usually, the mechanics are handled by a statistics program or calculator,
but it’s good to know the formulas.
REASONING OF HYPOTHESIS
TESTING (CONT.)
3. Mechanics (continued)
•
The ultimate goal of the calculation is to obtain a P-value.
•
The P-value is the probability that the observed statistic value (or
an even more extreme value) could occur if the null model were
correct.
•
If the P-value is small enough, we’ll reject the null hypothesis.
•
Note: The P-value is a conditional probability—it’s the probability
that the observed results could have happened if the null hypothesis
is true.
REASONING OF HYPOTHESIS
TESTING (CONT.)
4. Conclusion
•
The conclusion in a hypothesis test is always a statement about
the null hypothesis.
•
The conclusion must state either that we reject or that we fail to
reject the null hypothesis.
•
And, as always, the conclusion should be stated in context.
STEPS FOR HYPOTHESIS TESTING
1. Check Conditions and show that you have checked these!
• Random Sample: Can we assume this?
• 10% Condition: Do you believe that your sample size is less than
10% of the population size?
• Nearly Normal:
• If you have raw data, graph a histogram to check to see if it is
approximately symmetric and sketch the histogram on your
paper.
• If you do not have raw data, check to see if the problem states
that the distribution is approximately Normal.
STEPS FOR HYPOTHESIS TESTING
(CONT.)
2. State the test you are about to conduct
Ex) One-Sample t-Test for Means
3. Set up your hypotheses
H 0:
H A:
4. Calculate your test statistic
𝒕𝒏−𝟏 =
𝒙−𝝁𝟎
𝒔
𝒏
5. Draw a picture of your desired area under the t-model, and
calculate your P-value.
STEPS FOR HYPOTHESIS TESTING
(CONT.)
6. Make your conclusion.
When your P-value is small enough (or below α, if given), reject
the null hypothesis.
When your P-value is not small enough, fail to reject the null
hypothesis.
CALCULATOR TIPS
Given a set of data:
• Enter data into L1
• Set up STATPLOT to create a histogram to check the nearly Normal condition
• STAT  TESTS  2:T-Test
• Choose Stored Data, then specify your data list (usually L1)
• Enter the mean of the null model and indicate where the data are (>, <, or ≠)
Given sample mean and standard deviation:
• STAT  TESTS  2:T-Test
• Choose Stats  enter
• Specify the hypothesized mean and sample statistics
• Specify the tail (>, <, or ≠)
• Calculate
EXAMPLE 1
A company has set a goal of developing a battery that lasts over 5 hours (300
minutes) in continuous use. A first test of 12 of these batteries measured the
following lifespans (in minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326,
311, and 288. Is there evidence that the company has met its goal?
EXAMPLE 1 (CONTINUED)
Find a 90% confidence interval for the mean lifespan of this type of battery.
EXAMPLE 2 (PARTNERS)
Cola makers test new recipes for loss of sweetness during storage. Trained tasters
rate the sweetness before and after storage. Here are the sweetness losses
(sweetness before storage minus sweetness after storage) found by 10 tasters for
one new cola recipe:
Are these data good evidence that the cola lost sweetness?
DAY 3
EXAMPLE 3
Psychology experiments sometimes involve testing the ability of rats to navigate
mazes. The mazes are classified according to difficulty, as measured by the mean
length of time it takes rats to find the food at the end. One researcher needs a
maze that will take the rats an average of about one minutes to solve. He tests one
maze on several rats, collecting the data shown. Test the hypothesis that the mean
completion time for this maze is 60 seconds at an alpha level of 0.05. What is your
conclusion?
38.4
57.6
46.2
55.5
62.5
49.5
38.0
40.9
62.8
44.3
33.9
93.8
50.4
47.9
35.0
69.2
52.8
46.2
60.1
56.3
55.1
CONFIDENCE INTERVALS &
HYPOTHESIS TESTS
• Confidence intervals and hypothesis tests are built from the same
calculations.
• They have the same assumptions and conditions.
• You can approximate a hypothesis test by examining a confidence
interval.
• Just ask whether the null hypothesis value is consistent with a
confidence interval for the parameter at the corresponding
confidence level.
CONFIDENCE INTERVALS &
HYPOTHESIS TESTS (CONT.)
• Because confidence intervals are two-sided, they correspond to two-sided
tests.
• In general, a confidence interval with a confidence level of C%
corresponds to a two-sided hypothesis test with an -level of
100 – C%.
• The relationship between confidence intervals and one-sided hypothesis
tests is a little more complicated.
• A confidence interval with a confidence level of C% corresponds to a
one-sided hypothesis test with an -level of ½(100 – C)%.
MAKING ERRORS
•
Here’s some shocking news for you: nobody’s perfect. Even
with lots of evidence we can still make the wrong decision.
•
When we perform a hypothesis test, we can make mistakes in
two ways:
I.
The null hypothesis is true, but we mistakenly reject it.
(Type I error)
II.
The null hypothesis is false, but we fail to reject it.
(Type II error)
MAKING ERRORS (CONT.)
• Which type of error is more serious depends on the situation at
hand. In other words, the importance of the error is context
dependent.
• Here’s an illustration of the four situations in a hypothesis test:
MAKING ERRORS (CONT.)
• http://www.youtube.com/watch?v=Q7fZXEW4mpA
• What type of error was made?
• How about OJ Simpson?
MAKING ERRORS (CONT.)
• How often will a Type I error occur?
• A Type I error is rejecting a true null hypothesis. To reject the
null hypothesis, the P-value must fall below . Therefore, when
the null is true, that happens exactly with a probability of . Thus,
the probability of a Type I error is our  level.
• When H0 is false and we reject it, we have done the right thing.
• A test’s ability to detect a false null hypothesis is called the power
of the test.
MAKING ERRORS (CONT.)
• When H0 is false and we fail to reject it, we have made a Type II
error.
• We assign the letter  to the probability of this mistake.
• It’s harder to assess the value of  because we don’t know what
the value of the parameter really is.
• When the null hypothesis is true, it specifies a single parameter
value, H0: parameter = hypothesized value.
• When the null hypothesis is false, we do not have a specific
parameter; we have many possible values.
• There is no single value for  --we can think of a whole collection
of ’s, one for each incorrect parameter value.
MAKING ERRORS (CONT.)
• One way to focus our attention on a particular  is to think about
the effect size.
• Ask “How big a difference would matter?”
• We could reduce  for all alternative parameter values by increasing
.
• This would reduce  but increase the chance of a Type I error.
• This tension between Type I and Type II errors is inevitable.
• The only way to reduce both types of errors is to collect more
data. Otherwise, we just wind up trading off one kind of error
against the other.
POWER OF THE TEST
• The power of a test is the probability that it correctly rejects a false null
hypothesis.
• The power of a test is 1 –  ; because  is the probability that a test fails
to reject a false null hypothesis and power is the probability that it does
reject.
• Whenever a study fails to reject its null hypothesis, the test’s power
comes into question.
• When we calculate power, we imagine that the null hypothesis is false.
POWER OF THE TEST (CONT.)
• The value of the power depends on how far the truth lies from the
null hypothesis value.
• The distance between the null hypothesis value, 0 , and the
truth,  , is called the effect size.
• Power depends directly on effect size. It is easier to see larger
effects, so the farther  is from 0, the greater the power.
ASSIGNMENTS
• Day 1: 7.1-7.4-Set A Book Page (Same as 6.5 B Book Page) # 1 - 5
7.1-7.4-Set B Book Page (Same as 6.1 – 6.4 Book Page) # 23, 24
• Day 2: 7.1-7.4-Set B Book Page (Same as 6.1 – 6.4 Book Page) # 1cd, 2cd,
29, 30, 33, 35
• Day 3: 7.1-7.4-Set B Book Page (Same as 6.1 – 6.4 Book Page) # 22, 25 –
28, 34