Transcript Hypothesis Testing

Hypothesis Testing
A hypothesis is a conjecture about a population.
Typically, these hypotheses will be stated in terms of a
parameter such as m (mean) or p (proportion).
A test of hypothesis is a statistical procedure used to
make a decision about the conjectured value of a
parameter.
We will make our decision based on observed values of
a test statistic and resulting p-value.
The Hypotheses
There are two hypotheses which we are comparing.
The null hypothesis, H0, specifies a value of a
parameter. This hypothesis is assumed to be true, and an
experiment is conducted to determine if the collected
data is contradictory to the null hypothesis.
The alternative (or research) hypothesis, Ha, gives an
opposing statement about the value of the parameter.
The collected data will be analyzed to see if it supports
the alternative hypothesis.
Statistical Terminology
Once the data is collected, we seek an answer to the
question: “If the null hypothesis is true, how likely are
we to observe this type of data, or data which is more
extreme in the direction of the alternative hypothesis?”
The observed significance level, or p-value of a test of
hypothesis is the probability of obtaining the observed
value of the sample statistic, or one which is even more
supportive of the alternative hypothesis, under the
assumption that the null hypothesis is true.
Empirical Rule Example
One thing you might think of initially is to see if the
value (or statistic’s value) is where you expect it to be.
If H0 says the population mean should be 78 bpm (heart
rate) with a standard error of the mean of 3 bpm, then
you know from the empirical rule that roughly 95% of
heart rates should lie between 72 bpm and 84 bpm.
So if a sample mean of patients on a new drug, for
example, have a rate of 73 bpm, that is expected. If the
sample mean is 86 bpm, that might be cause for concern.
Test Statistics and P-values
Obviously, we must be able to calculate these p-values.
A test statistic is a quantity calculated from the sampled
value of a statistic, which is then used to calculate a
p-value for a test of hypothesis.
We recall that under appropriate conditions, we know
the sampling distribution of the sample mean follows a
normal distribution, and we can find a t-score using
M m
t
 s



n

Using P-values to make Decisions
We must decide when to accept that the research
hypothesis has been proven beyond some reasonable
doubt. This is called a significance level, and is denoted
by a.
When the probability of the sample data is below this
level of significance, we conclude there is enough
evidence to assert the research hypothesis is true.
So we accept Ha when the p-value is less than or equal to
the level of significance. (decision rule)
Steps of the t-Test of Hypothesis for m
1. Identify the null and alternative hypotheses
2. Give the decision rule
3. Identify the form of the test statistic and calculate it
based on the sampled data
4. Verify the method is valid (random sample, s is
unknown, and n is at least 30 or population is known
to be normally distributed )
5. Calculate the p-value based on the t-score and Ha
6. Give an interpretation. (sentence)
Hypothesis Testing Example
A local radio station claims that they play an average of
54 minutes every hour. A listener believes this
statement is incorrect, and wants to prove that this
station plays less than 54 minutes of music, on
average, each hour. The listener monitors 36
different 1-hour time slots and records the amount
of time during that hour which the station played
music. She finds that the mean and standard
deviation for the 36 observations are 53.2 minutes
and 2.1 minutes, respectively. Does this provide
evidence that the listener is correct? Test using a
.05 level of significance.
Hypothesis Testing Example
1. H0: m = 54 minutes
Ha: m < 54 minutes
2. Decision Rule: Accept Ha if the p-value < .05.
3.
M m
53.2  54  0.8
t


= – 2.3
 s
  2.1
 0.35

 

n 
36 

4. Valid since a random sample was taken, s is
unknown so we use s = 2.1 min, and
n = 36 is “at least 30”.
Hypothesis Testing Example
5. p-value = .0138
df = n – 1 = 36 – 1 = 35,
We will obtain the p-value from software using the
actual sample data set.
6. At the .05 level of significance, there is sufficient
evidence to conclude the mean time per hour this
station plays music is less than 54 minutes.
Hypothesis Testing Example
Historically, the mean time until a patient feels relief
from pain after surgery on the standard medication has
been 3 minutes. A new drug is available which
researchers hope will lead to faster average relief times
for patients after surgery. A random sample of 41
patients were given this new medication after surgery
and the time until the patient felt relief was recorded.
What is the research hypothesis?
Hypothesis Testing Example
A new drug is available which researchers hope will
lead to faster average relief times for patients after
surgery.
What is the research hypothesis?
So Ha: m < 3 minutes where m is the mean relief time
among all patients who take this new drug after
surgery.
Also, this implies H0: m > 3 minutes.
Hypothesis Testing Example
Suppose the mean relief time was 2.8 minutes and the
standard deviation was 0.92 minutes for the 41 patients.
Does this allow the researchers to conclude the new
drug is faster, on average? Use a = 0.05.
Decision rule: Accept Ha if the p-value < 0.05.
Test Statistic: t = -1.39 (see next slide for output)
P-value: 0.086
At the .05 level of significance, there is not enough
evidence to conclude the new drug is faster, on average.
Hypothesis Testing Example
StatCrunch Output for the Entire Test:
Hypothesis test results:
μ : population mean
H0 : μ = 3
HA : μ < 3
Mean
μ
Sample Mean
2.8
Std. Err.
0.14367986
DF
40
T-Stat
-1.3919835
P-value
0.0858
Possible Errors
Realize, that a small p-value (or observed level of
significance) suggests that the alternative hypothesis
is true, but does not guarantee it is true.
A Type I Error consists of concluding that the
alternative hypothesis is true when, in fact, the null
hypothesis is true.
A Type II Error consists of concluding that the null
hypothesis is true when, in fact, the alternative
hypothesis is true.
Type I or Type II Errors
We can examine each error and attempt to control how
willing we are to commit each.
The probability of committing a Type I Error is given by
a(alpha), the level of significance of the test.
The probability of committing a Type II Error is given
by b (beta).
We will decide the seriousness of each error, and then
determine relative values for a and b.
Type I or Type II Errors
H0: Dental lab is not contaminated with mercury vapor.
Ha: Dental lab is contaminated with mercury vapor.
A Type I Error occurs if we conclude the lab is
contaminated when, in fact, the lab is not
contaminated with mercury vapor.
Consequences: The office would probably be closed,
and procedures to reduce the mercury vapor levels
would be implemented. This means costs for
decontamination and lack of income from patients,
but no one is endangered.
Type I or Type II Errors
H0: Lab is not contaminated with mercury vapor.
Ha: Lab is contaminated with mercury vapor.
A Type II Error occurs if we conclude the lab is NOT
contaminated when, in fact, the lab is contaminated
with mercury vapor.
Consequences: The office would be open and new
patients and workers could be exposed to high levels
of mercury vapor. This means possible law suits,
people getting sick, dangerous work environment. It
could lead to a bad reputation for this dental office.
Type I or Type II Errors
H0: Lab is not contaminated with mercury vapor.
Ha: Lab is contaminated with mercury vapor.
Which error is worse? Probably Type II in my opinion.
Hence, I want to avoid this, so I set the probability
low. The Type I Error means a loss of income, so
it’s not good for business, but it’s not as serious.
I’d set a more than b.
Type I or Type II Errors
Be able to define a Type I Error and a Type II error in
terms of a scenario.
Be able to discuss the consequences of each type of
error.
Set relative values for a and b depending on those
consequences (which is larger and which is smaller,
or are both about equal?).
There are no rules on one error being worse than the
other in all situations.