Population:
The set of all individuals of interest (e.g. all women, all college students)
Sample:
A subset of individuals selected from the population from whom data is collected
probability
Probability
What we learned from Probability
1) The mean of a sample can be treated as a random variable.
3) Because of this, we can find the probability that a given population might randomly produce a particular range of sample means.
)()( somethingZPsomethingXP Use table E.10
X nX
2) By the central limit theorem, sample means will have a normal
distribution (for n > 30) with and
Population:
The set of all individuals of interest (e.g. all women, all college students)
Sample:
A subset of individuals selected from the population from whom data is collected
Inferential statistics
Inferential statistics
Once we’ve got our sample
The key question in statistical inference:
Could random chance alone have produced a sample like ours?
Once we’ve got our sample
Distinguishing between 2 interpretations of patterns in the data:
Random Causes:
Fluctuations of chance
Systematic Causes Plus Random Causes:
True differences in the population
Bias in the design of the study
Inferential statistics separates
Reasoning of hypothesis testing
1. Make a statement (the null hypothesis) about some unknown population parameter.
2. Collect some data.
3. Assuming the null hypothesis is true, what is the probability of obtaining data such as ours? (this is the “p-value”).
4. If this probability is small, then reject the null hypothesis.
Step 1: Stating hypotheses
Null hypothesis– H0
– Straw man: “Nothing interesting is happening”
Alternative hypothesis– Ha
– What a researcher thinks is happening– May be one- or two-sided
Hypotheses are in terms of population parameters
One-sided Two-sided
H0: µ=110 H0: µ = 110
H1: µ < 110 H1: µ ≠ 110
Step 1: Stating hypotheses
Step 2: Set decision criterion
• Decide what p-value would be “too unlikely”
• This threshold is called the alpha level.
• When a sample statistic surpasses this level, the result is said to be significant.
• Typical alpha levels are .05 and .01.
More on setting a criterion
• The retention region. The range of sample mean values that are “likely” if H0 is true.
If your sample mean is in this region, retain the null hypothesis.
• The rejection region.The range of sample mean values that are “unlikely” if H0 is true.
If your sample mean is in this region, reject the null hypothesis
Accept H0
Reject H0 Reject H0
Zcrit Zcrit
Setting a criterion
0H
Null distribution
Step 3: Compute sample statistics
A test statistic (e.g. Ztest, Ttest, or Ftest) is information we get from the sample that we use to make the decision to reject or keep the null hypothesis.
A test statistic converts the original measurement (e.g. a sample mean) into units of the null distribution (e.g. a z-score), so that we can look up probabilities in a table.
Accept H0
Reject H0 Reject H0
Zcrit Zcrit
Test Statistics
hyp
Null distribution
Ztest?
• If we want to know where our sample mean lies in the null distribution, we convert X-bar to our test statistic Ztest
• If an observed sample mean were lower than z=-1.65 then it would be in a critical region where it was more extreme than than 95% of all sample means that might be drawn from that population
Accept H0
Reject H0 Reject H0
Zcrit Zcrit
Step 4: Make a decision
If our sample mean turns out to be extremely unlikely under the null distribution, maybe we should revise our notion of µH0
We never really “accept” the null. We either reject it, or fail to reject it.
Steps of hypothesis testing
1. State hypothesis (H0, HA)
2. Select a criterion (alpha, Zcrit)
3. Compute test statistic (Ztest) and get a p-value
4. Make a decision
Z as test statistic
X
Htest
XZ
0
• Z test-statistic converts a sample mean into a z-score from the null distribution.
•Zcrit is the criterion value of Z that defines the rejection region
•Ztest is the value of Z that represents the sample mean you calculated from your data
Standard error!!!!
• p-value is the probability of getting a Ztest as extreme as yours under the null distribution
Z as test statistic
X
hypcalc
XZ
• All test statistics are fundamentally a comparison between what you got and what you’d expect to get from chance alone
Deviation you got
Deviation from chance alone
• If the numerator is considerably bigger than the denominator, you have evidence for a systematic factor on top of random chance
Example I
Tim believes that his “true weight” is 187 lbs with a standard deviation of 3 lbs.
Tim weighs himself once a week for four weeks. The average of these four measurements is 190.5.
Are the data consistent with Tim’s belief?
Example I
33.243
1875.1900
X
Htest
XZ
1. H0: = 187 HA: > 187
2. Criterion? Let’s say alpha=.05. That would be Zcrit = 1.65
3. An X-bar of 190.5 is what Ztest? What is the probability of getting a Ztest as high as ours?
4. If H0 were true, there would be only about a 1% chance of randomly obtaining the data we have. Reject H0.
0099.)33.2( ZP
x = 187190.5x= 1.5
z = 190.5-187 = 2.33 3
4
0 2.33
Example I illustrated
1.65
Reject H0
Zcrit
0.01
Ztest
Exercise
We have a sample of 500 students whose average score on some standardized test is 461. We think they are a particularly gifted bunch.
Assume the general student population has a distribution of scores that is approximately normal with µ = 450 and = 100.
Does our sample come from a population with a mean of 450? Or are they a better test-taking species?
H0: µ = 450H1: µ > 450
Exercise
How to proceed?
Let’s:- Select a criterion - Calculate a z-score- Compare our sample z with our criterion- Make a decision
Exercise
We have a sample of 500 students whose average score on some standardized test is 461. We think they are a particularly gifted bunch.
Assume the general student population has a distribution of scores that is approximately normal with µ = 450 and = 100.
Does our sample come from a population with a mean of 450? Or are they a better test-taking species?
H0: µ = 450H1: µ > 450
We reject the null hypothesis because sample means of 461 or larger have a very small probability. (We expect such large means less than 1% of the time.)
Exercise illustrated
When we reject a null hypothesis, it is because
(a) if we believe the null hypothesis, there is only a small probability of getting data like ours by chance alone.
(b) if we believe our data, and don’t think it came from an unlikely chance event, the null distribution is probably not true.
• If HA states is < some value, critical region occupies left tail
• If HA states is > some value, critical region occupies right tail
One-tailed tests
H0: µ = 100
H1: µ > 100
Values that differ “significantly”
from 100100
Points Right
Fail to reject H0 Reject H0
Right-tailed tests
alpha
Zcrit
H0: µ = 100
H1: µ < 100
100
Values that differ “significantly”
from 100
Points Left
Fail to reject H0Reject H0
Left-tailed tests
alpha
Zcrit
One- vs. two-tailed tests
• In theory, should use one-tailed when 1. Change in opposite direction would be meaningless2. Change in opposite direction would be uninteresting3. No rival theory predicts change in opposite direction
• By convention/default in the social sciences, two-tailed is standard
• Why? Because it is a more stringent criterion (as we will see). A more conservative test.
• HA is that µ is either greater or less than µH0
HA: µ ≠ µH0
is divided equally between the two tails of the critical region
Two-tailed hypothesis testing
H0: µ = 100
H1: µ 100
100
Values that differ significantly from 100
Means less than or greater than
Fail to reject H0Reject H0 Reject H0
Two-tailed hypothesis testing
alpha
Zcrit Zcrit
100
Values that differ “significantly”
from 100
Fail to reject H0Reject H0
One tail
.05
Zcrit
100
Values that differ significantly from 100
Fail to reject H0Reject H0 Reject H0Two tail
.025 .025
Zcrit Zcrit
Example
We have a sample of 36 children of geniuses. They have an average IQ of 110. We want to know whether they are significantly different from the general population of children, who have µ=100 and σ=25.
1. Test the hypothesis that the mean of this group is higher than that of the population.
2. What is Ztest?
3. What is Zcrit for alpha = .05? For alpha = .01? Do we reject the null for either case?
4. What is the exact p-value for this test?
Example
• Ztest= 10/4.16 = 2.4
• alpha .05, Zcrit=1.64;
• alpha .01, Zcrit=2.33
Reject Ho
Ztest
• P(Z>2.4)=.008
Example
We have a sample of 36 children of geniuses. They have an average IQ of 110. We want to know whether they are significantly different from the general population of children, who have µ=100 and σ=25.
1. Test the hypothesis that the mean of this group is not equal to that of the population
2. What is Ztest?
3. What is Zcrit for alpha = .05? For alpha = .01? Do we reject the null for either case?
4. What is the exact p-value for this test?
Example
• Ztest= 10/4.16 = 2.4
• alpha .05, Zcrit=1.96;
• alpha .01, Zcrit=2.58
Reject Ho
Ztest
• P(/Z/>2.4)=.016