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Lecture 10
Hypothesis Testing
A hypothesis is a conjecture about the distribution of some random
variables. For example, a claim about the value of a parameter of
the statistical model.
There are two types of hypotheses:
The null hypothesis, , is the current belief.
The alternative hypothesis,
want to show.
, is your belief; it is what you
Examples: Each of the following situations requires a significance
test about a population mean. State the appropriate null hypothesis
and alternative hypothesis
in each case.
(a) The mean area of the several thousand apartments in a new
development is advertised to be 1250 square feet. A tenant
group thinks that the apartments are smaller than advertised.
They hire an engineer to measure a sample of apartments to
test their suspicion.
(b) Larry's car consume on average 32 miles per gallon on the
highway. He now switches to a new motor oil that is
advertised as increasing gas mileage. After driving 3000
highway miles with the new oil, he wants to determine if his
gas mileage actually has increased.
(c) The diameter of a spindle in a small motor is supposed to be
5 millimeters. If the spindle is either too small or too large,
the motor will not perform properly. The manufacturer
measures the diameter in a sample of motors to determine
whether the mean diameter has moved away from the target.
Guidelines for Hypothesis testing
Hypothesis testing is a proof by contradiction. The testing process
has four steps:
Step 1: Assume
is true.
Step 2: Use statistical theory to make a statistic (function of the
data) that includes . This statistic is called the test statistic.
Step 3: Find the probability that the test statistic would take a
value as extreme or more extreme than that actually observed.
Think of this as: probability of getting our sample assuming
is
true.
Step 4: If the probability we calculated in step 3 is high it means
that the sample is likely under
and so we have no evidence
against . If the probability is low, there are two possibilities:
- we observed a very unusual event, or
- our assumption is wrong
Test Statistic
• The test is based on a statistic that estimates the parameter
that appears in the hypotheses. Usually this is the same
estimate we would use in a confidence interval for the
parameter. When
is true, we expect the estimate to take a
value near the parameter value specified in
.
• Values of the estimate far from the parameter value specified by
give evidence against
. The alternative hypothesis
determines which directions count against .
• A test statistic measures compatibility between the null
hypothesis and the data.
• To assess how far the estimate is from the parameter, standardize
the estimate. In many common situations the test statistics has the
form
Example: An air freight company wishes to test whether or not the
mean weight of parcels shipped on a particular root exceeds 10
pounds. A random sample of 49 shipping orders was examined and
found to have average weight of 11 pounds. Assume that the
standard deviation of the weights is 2.8 pounds.
Solution:
Graphical Representation
Suppose we want to test a set of hypotheses concerning a
parameter based on a random sample
.
vs
̂ is the estimate of our parameter .
Rejection Region (RR) is the specified values of the test statistics
for which we reject .
The probability that defines the critical region is called the size of
the test or level of the significance of the test and is denoted by α.
Example: The hourly wages in a particular industry are normally
distributed with mean $13.20 and standard deviation $2.50. A
company employs 40 workers paying them an average of $12.20
per hour. Can this company be accused of paying substandard
wages? Use
.
Solution:
Decision Errors
When we perform a statistical test we hope that our decision will
be correct, but sometimes it will be wrong. There are two possible
errors that can be made in hypothesis test.
Definition: The error made by rejecting the null hypothesis
when in fact
is true is called a type I error.
The error made by failing to reject the null hypothesis
when in
fact
is false is called a type II error.
Note: The level of significance of the test is also the probability of
type I error, denoted by , i.e.
The probability of a type II error is denoted by .
Example: An experimenter has prepared a drug dosage level that
she claims will induce sleep for 80% of people suffering from
insomnia. In an attempt to disprove her claim, we administer her
prescribed dosage to 20 insomniacs and observe X, the number of
people for whom the drug dose induces sleep. We wish to test
vs
. Assume
.
P-value
Definition: The probability, assuming
is true, that the test
statistic would take a value as extreme or more extreme than that
actually observed is called the P-value of the test. The smaller the
P-value, the stronger the evidence against
provided by the data.
Guideline for how small is “small”:
P-value > 0.1 provides no evidence against
.
0.05 < P-value < 0.1 provides weak evidence against
.
0.01 < P-value < 0.05 provides moderated evidence against
P-value < 0.01 provides strong evidence against
.
.
We can compare the P-value we calculate with a fixed value that
we regard as decisive. The decisive value of P is called the
significance level (this is our ). Most common values for are
0.1, 0.05, 0.01.
If the P-value is as small or smaller than , we say that the data are
statistically significant at level . In other words, the P-value is
the smallest level of significance for which the null hypothesis
should be rejected.
Example: 85% of the general public is right-handed. A survey of
300 chief executive officers of large corporations found that 95%
were right-handed. Is this difference in percentages statistically
significant? Use
. Find the P-value for the test.
Solution:
Tests for a Population Mean ( is known)
where
is the specified value of .
Example: In 1999, it was reported that the mean serum cholesterol
level for female undergraduates was 168 mg/dl with a standard
deviation of 27 mg/dl. A recent study at Baylor University
investigated the lipid levels in a cohort of sedentary university
students. The mean total cholesterol level among n = 71 females
was ̅
. Is this evidence that cholesterol levels of sedentary
students differ from the previously reported average?
Solution:
Tests for a Population Mean ( is unknown)
Recall: (one-sample t CI)
Example: Founded in 1998, Telephia provides a wide variety of
information on cellular phone use. In 2006, Telaphia reported that,
on average, United Kingdom (U.K.) subscribers with thirdgeneration technology (3G) phones spent an average of 8.3 hours
per month listening to full-track music on their cell phones.
Suppose we want to determine a 95% CI for the U.S. average and
draw the following random sample of size 8 from the U.S.
population of 3G subscribers:
5 6 0 4 11 9 2 3
The sample mean is ̅
and the standard deviation s = 3.63 with
degrees of freedom n - 1 = 7.
Example: Suppose that, for the U.S. data in example before we
want to test whether the U.S. average is different from the reported
U.K. average.
Power
The ability of a test to detect that
is false is measured by the
probability that the test will reject
when an alternative is true.
The higher this probability is, the more sensitive the test is.
Definition: The probability that a fixed size
test will reject
when
is false is called the power of the test.
A powerful test has a large probability of rejecting
false.
when it is
Example: Can a 6-month exercise program increase the total body
bone mineral content (TBBMC) of young women? A team of
researchers is planning a study to examine this question. Based on
the results of a previous study, they are willing to assume that
for the percent change in TBBMC over the 6-month period.
A change in TBBMC of 1% would be considered important, and
the researchers would like to have a reasonable chance of detecting
a change this large or larger. Is 25 subjects a large enough sample
for this project?
Three steps to find the power of the test:
1. State , , the particular alternative we want to detect, and
the significance level .
2. Find the values of ̅ (or other estimates) that will lead to reject
.
3. Calculate the probability of observing these values of ̅ when
the alternative is true.
How to increase the power?
Back to Error Probabilities
Example: The mean outer diameter of a skateboard bearing is
supposed to be 22.000 millimeters (mm). The outer diameters vary
Normally with standard deviation
mm. When a lot of
bearings arrives, the skateboard manufacturer takes an SRS of 5
bearings from the lot and measures their outer diameters. The
manufacturer rejects the bearings if the sample mean diameter is
significantly different from 22 at the 5% significance level.
Suppose the producer and the manufacturer agree that a lot of
bearings with mean 0.015 mm away from 22 should be rejected.
Significance and Type I error: The significance level
of any
fixed level test is the probability of a Type I error. That is, is the
probability that the test will reject
when
is in fact true.
Power and Type II error: The power of a fixed level test to detect a
particular alternative is 1 minus the probability of a Type II error
for that alternative.
Comparing Two Means
Assume we have two populations of interest, each with unknown
mean . Choose an SRS of size from one normal population
having mean
and standard deviation
and an independent
SRS of size
from another normal population having mean
and standard deviation
. The estimate of the difference in the
population means is
̂
̂
where ̅ and ̅ are sample means.
Distribution of ̅
̅ :
̂
̅
̅
Example: A fourth-grade class has 12 girls and 8 boys. The
children’s heights are recorded on their 10th birthdays. Based on
information from the National Health and Nutrition Examination
Survey, the heights (in inches) of 10-year-old girls are
distributed Normally with mean 56.8 and standard deviation 2.7
and the heights (in inches) of 10-year-old boys are distributed
Normally with mean 55.7 and standard deviation 3.8. Assume
that the heights of the students in the class are random samples
from the populations. What is the probability that the girls’
average height is greater than the boys’ average height?
Solution:
Here we know
and
, which is quite rare.
So in general, there are two ways to compare the means of two
normal populations. This is due to the fact that there are two
distinct possibilities:
1.
2.
and
and
are unknown and equal.
are unknown and unequal.
Comparing Two Mean: Variances Unequal
Assume
and
are unknown. We estimate them by
and
.
Example: An educator believes that new directed reading activities
in the classroom will help elementary school pupils improve some
aspects of their reading ability. She arranges for a third-grade class
of 21 students to take part in these activities for an eight-week
period. A control classroom of 23 third-graders follows the same
curriculum without the activities. At the end of the eight weeks, all
students are given a Degree of Reading Power (DRP) test, which
measures the aspects of reading ability that the treatment is
designed to improve. The data appear in the table below:
Two-Sample t CI:
Choose an SRS of size
from a Normal population with
unknown mean
and an independent SRS of size
from
another Normal population with unknown mean .
A
(
)
CI for
( ̅
is given by
̅ )
√
where
is the value for
density curve with area
between
and
. The value of the degrees of freedom k is
approximated by software or we use the smaller of
and
.
Example: How much improvement?
Comparing Two Means: Variances Equal (Pooled Test)
Suppose we have two Normal populations with the same
variances:
, is unknown.
The pooled two-sample t procedures:
Choose an SRS of size
from a Normal population with
unknown mean
and an independent SRS of size
from
another Normal population with unknown mean .
A
(
)
CI for
( ̅
where
between
is given by
√
̅ )
is the value for
and
.
density curve with area
To test the hypothesis
sample t statistic
, compute the pooled two̅
̅
√
In terms of a random variable T having the
the P-value for a test of
against
distribution,
Example: Does increasing the amount of calcium in our diet reduce
blood pressure? Examination of a large sample of people revealed
a relationship between calcium intake and blood pressure, but such
observational studies do not establish causation. A randomized
comparative experiment gave one group of 10 people a calcium
supplement for 12 weeks. The control group of 11 people received
a placebo that appeared identical. Table below gives the seated
systolic blood pressure for all subjects at the beginning and end of
12-week period, in millimeters of mercury. The table also shows
the decrease of each subject. An increase appears as a negative
entry.
Does increase calcium reduce blood pressure?
How different are the calcium and placebo groups?