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