Download Hypothesis Testing 2

Sociology 5811:
Lecture 10: Hypothesis Tests
Copyright © 2005 by Evan Schofer
Do not copy or distribute without
permission
Announcements
• Problem Set #3 Due next week
• Problem set posted on course website
• We are a bit ahead of reading assignments in
Knoke book
• Try to keep up; read ahead if necessary
Z-scores for Sampling Distributions
• New application of Z-Scores:
(Yi  Y )
(Y  μ)
Zi 

sY
σY
• “Old” formula is for variable distributions:
• It calculates the # standard deviations a case falls from Y-bar
• New formula is for sampling distributions
• It tells you the number of standard errors Y-bar falls from the
population mean 
• We can also compute distance from Y-bar to a hypothetical
value of  (as we did last class).
Hypothesis Testing
• Hypothesis Testing:
• A formal language and method for examining
claims using inferential statistics
• Designed for use with probabilistic empirical assessments
• Because of the probabilistic nature of inferential
statistics, we cannot draw conclusions with
absolute certainty
• We cannot “prove” our claims are “true”
• While it is improbable, we will occasionally draw a sample
that is highly unusual, leading to incorrect conclusions.
Hypothesis Testing
• The logic of hypothesis testing:
• We cannot “prove” anything
• Instead, we will cast doubt on other claims, thus
indirectly supporting our own
• Strategy:
– 1. We first state an “opposing” claim
• The opposite of what we want to claim
– 2. If we can cast sufficient doubt on it, we are forced
(grudgingly) to accept our own claim.
Hypothesis Testing
• Example: Suppose we wish to argue that our
school is above the national standard
• First we state the opposite:
• “Our school is not above the national standard”
• Next we state our alternative:
• “Our school is above the national standard”
• If our statistical analysis shows that the first claim
is highly improbable, we can “reject” it, in favor
of the second claim
• …“accepting” the claim that our school is doing well.
Hypothesis Testing: Jargon
• Hypotheses: Claims we wish to test
• Typically, these are stated in a manner specific
enough to test directly with statistical tools
– We typically do not test hypotheses such as “Marx
was right” / “Marx was wrong”
– Rather: The mean years of education for Americans
is/is not above 18 years.
Hypothesis Testing: Jargon
• The hypothesis we hope to find support for is
referred to as the alternate hypothesis
• The hypothesis counter to our argument is
referred to as the null hypothesis
• Null and alternative hypotheses are denoted as:
• H0: School does not exceed the national standard
• H-zero indicates null hypothesis
• H1: School does exceed national standard
• H-1 indicates alternate hypotheses
• Sometimes called: “Ha”
Hypothesis Testing: More Jargon
• If evidence suggests that the null hypothesis is
highly improbable, we “reject” it
• And, we “accept” the alternative hypothesis
• So, typically we:
• Reject H0, accept H1
• Or:
• Fail to reject H0, do not find support for H1
• That was what happened when we “tested” whether our
school exceeded the national standard (=60).
Hypothesis Testing
• In order to conduct a test to evaluate hypotheses,
we need two things:
• 1. A statistical test which reflects on the
probability of H0 being true rather than H1
• Here, we used a z-score/t-score to determine the probability
of H0 being true
• 2. A pre-determined level of probability below
which we feel safe in rejecting H0 (a)
• In the example, we wanted to be 95% confident… a =.05
• But, the probability was .105, so we couldn’t conclude that
the school met the national standard!
Hypothesis Test for the Mean
• Example: Corporate Salaries
– Imagine I’m a human resources director of “Evan.com”
• Our engineers are paid 50,000/year
• I suspect that our salaries are not competitive
– So, I survey employees of our main competitor…
• I sample 20 people and observe a mean salary of 55K
– Y-bar is 55K, but we don’t know  …
• Issue: Are our salaries below the industry?
– Hypotheses:
• H0: Competitor’s salaries are no better ( <= 50K)
• H1: Competitors salaries are better ( > 50K).
Hypothesis Test: Example
• It looks like the other company pays more::
• Average Salary is 55K, compared to our baseline of 50K
• Question: Can we reject the null hypothesis and
accept the alternate hypothesis?
• Answer: No! It is possible that we just drew an
atypical sample.
• The true population mean for the competitor may be higher.
Hypothesis Test: Example
• We need to use our statistical knowledge to
determine:
• What is the probability of drawing a sample
(N=20) with mean of 55K from a population of
mean 50K?
• If that is a probable event, we can’t draw very strong
conclusions. It is likely that competitor salaries are the same.
• But, if the event is very improbable, we can conclude that the
competitor salaries exceed 50K.
Hypothesis Test: Example
• How would we determine the probability that the
competitor mean salary is really only 50K?
• Answer: We apply the Central Limit Theorem to
determine the shape of the sampling distribution
• And then calculate a Z-value or T-value based on it
• Suppose we chose an alpha (a) of .05
• If we observe a t-value with probability of only .0023, then
we can reject the null hypothesis.
• If we observe a t-value with probability of .361, we cannot
reject the null hypothesis.
Hypothesis Test: Steps
• 1. State the research hypothesis (“alternate
hypothesis), H1
• 2. State the null hypothesis, H0
• 3. Choose an a-level (alpha-level)
• Typically .05, sometimes .10 or .01
• 4. Look up value of test statistic corresponding to
the a-level (called the “critical value”)
• Example: find the “critical” t-value associated with a=.05
Hypothesis Test: Steps
• 5. Use statistics to calculate a relevant test
statistic.
• T-value or Z-value
• Soon we will learn additional ones
• 6. Compare test statistic to “critical value”
• If test statistic is greater, we reject H0
• If it is smaller, we cannot reject H0
Hypothesis Test: Steps
• Alternate steps:
• 3. Choose an alpha-level
• 4. Get software to conduct relevant statistical test
• Software will compute test statistic and provide a
probability… the probability of observing a test statistic of
a given size.
• If this is lower than alpha, reject H0
Hypothesis Test: Errors
• Due to the probabilistic nature of such tests, there
will be periodic errors.
– Sometimes the null hypothesis will be true, but we
will reject it
• When we falsely reject H0, it is called a Type I error
• Our alpha-level determines the probability of this
– Sometimes we do not reject the null hypothesis, even
though it is false
• When we falsely fail to reject H0, it is called a Type II error
– In general, we are most concerned about Type I
errors… we try to be conservative.
Hypothesis Tests About a Mean
• Possible hypothesis tests for a single mean:
• 1. Population mean is not equal to a certain value
• Null hypothesis is that the mean is equal to that value
• 2. Population mean is higher than a value
• Null hypothesis: mean is equal or less than a value
• 3. Population mean is lower than a value
• Null hypothesis: mean is equal or greater than a value
• We will learn more interesting kinds of tests:
• Tests comparing means of two groups
• Tests about correlations, regressions, etc.
Hypothesis Tests About Means
• Example: Bohrnstedt & Knoke, section 3.93, pp.
108-110. N = 1015, Y-bar = 2.91, s=1.45
• H0: Population mean  = 4
• H1: Population mean  ≠ 4
• Strategy:
• 1. Choose Alpha (let’s use .001)
• 2. Determine the Standard Error
• 3. Use S.E. to determine the probability of the
observed mean (Y-bar), IF the population mean
is really 4.
• 4. If the probability is below .001, reject H0
Example: Is  =4?
• Let’s determine how far Y-bar is from hypothetical =4
• In units of standard errors
(Y  μ) (2.91  4)  1.09
t


σY
σ̂ Y
σ̂ Y
sY
1.45
σ̂ Y 

 .046
N
1015
t  1.09 / .046  24.0
• Y-bar is 24 standard errors below 4.0!
Hypothesis Tests About a Mean
• A Z-table (if N is large) or a T-table will tell us
probabilities of Y-bar falling Z (or T) standard
deviations from 
• In this example, the desired a = .001
• Which corresponds to t=3.3 (taken from t-table)
– That is: .001 (i.e, .1%) of samples (of size 1015) fall
beyond 3.3 standard errors of the population mean
• 99.9% fall within 3.3 S.E.’s.
Hypothesis Tests About a Mean
• There are two ways to finish the “test”
• 1. Compare “critical t” to “observed t”
• Critical t is 3.3, observed t = -24
• We reject H0: t of -24 is HUGE, very improbable
• It is highly unlikely that  = 4
• 2. Actually calculate the probability of observing
a t-value of 24, compare to pre-determined a
• If observed probability is below a, reject H0
– In this case, probability of t=27 is .0000000000000…
• Very improbable. Reject H0!
Two-Tail Tests
• Visually: Most Y-bars should fall near 
• 99.9% CI: –3.3 < t < 3.3, or 3.85 to 4.15
Mean of 2.91
(t=24) is far
into the red
area (beyond
edge of graph)
Sampling Distribution
of the Mean
3.85
Z=-3.3
4
4.15
Z=+3.3
Hypothesis Tests About a Mean
• Note: This test was set up as a “two-tailed test”
• Hypothesis was  ≠ 4… It didn’t specify a direction
• Meaning, that we reject H0 if observed Y-bar falls in either
tail of the sampling distribution
• Not all tests are done that way… Sometimes you only want
to reject H0 if Y-bar falls in one particular tail.
Hypothesis Testing
• Definition: Two-tailed test: A hypothesis test in
which the a-area of interest falls in both tails of a
Z or T distribution.
• Example: H0: m = 4; H1: m ≠ 4
• Definition: One-tailed test: A hypothesis test in
which the a-area of interest falls in just one tail
of a Z or T distribution.
• Example: H0:  > or = 4; H1:  < 4
• Example: H0:  < or = 4; H1:  > 4
• This is called a “directional” hypothesis test.
Hypothesis Tests About Means
• A one-tailed test: H1:  < 4
• Entire a-area is on left, as opposed to half (a/2)
on each side. SO: the critical t-value changes.
4
Hypothesis Tests About Means
• T-value changes because the alpha area (e.g., 5%)
is all concentrated in one size of distribution,
rather than split half and half.
•
One tail
vs.
Two-tail:
a.05
a/2.025
a/2.025
Looking Up T-Tables
How much does the
95% t-value change
when you switch from
a 2-tailed to 1-tailed
test?
Two-tailed
test (20 df):
t=2.086
One-tailed
test (20df)
t=1.725
Hypothesis Tests About Means
• Use one-tailed tests when you have a directional
hypothesis
• e.g.,  > 5
• Otherwise, use 2-tailed tests
• Note: Switching to a one-tailed test lowers the
critical t-value needed to reject H0.
Tests for Differences in Means
• A more useful and interesting application of these
same ideas…
• Hypothesis tests about the means of two
different groups
• Up until now, we’ve focused on a single mean for a
homogeneous group
• It is more interesting to begin to compare groups
• Are they the same? Different?
• We’ll do that next class!