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More Statistics

Andrew MartinPS 372

University of Kentucky

Inference

Inference refers to reasoning from available information or facts to reach a conclusion. However, there is no guarantee the inference is correct.

In fact, inferences are sometimes incorrect.

Inference

In statistical inference the estimated values of unknown population parameters are sometimes incorrect.

Concerning hypothesis testing, there are two types of mistakes we can make: a Type I error and a Type II error.

Type I Error

Type I error occurs whenever one rejects a true null hypothesis.

Suppose:(1) in reality, the coin is fair (that is, H

0: P = .5)

(2) You decide to reject H0 if 0,1,9 or 10 heads occurs.

(3) your opponent obtains 9 heads in 10 tosses.(4) You reject the hypothesis and accuse the

person of being unfair.

Type II Error

Type II error occurs whenever one fails to reject a false null hypothesis.

Suppose:(1) in reality, the coin is unfair (that is, H

0: P = .9)

(2) You decide to reject H0 only if 10 heads occurs.

(3) Your opponent obtains 9 heads in 10 tosses.(4) You do not reject the null hypothesis (H

0: P = .5)

even though it is false.

What are the chances?

The probability of committing a type I error is the “size” of the critical region. It is designated by small Greek letter alpha (α).

The probability of committing a type II error (β) depends on :

1) how far the true value of the population parameter is from the hypothesized one

2) the sample size – the larger the sample size, the lower the probability of committing a type II error.

Standard Error

Imagine taking an endless number of independent samples of size N from a fixed population that has a mean of μ and a standard deviation σ.

For each sample, you calculate Y and the standard deviation

¿¿

σ

Standard Error

The standard deviation of the sampling distribution is called the standard error of the mean, or standard error.

Where is the sample standard deviation and N is the sample size.

σ

Standard Error

The expression implies that as the sample size gets larger and lagers, the standard error decreases in numerical value.

As a result, as the sample increases we expect Y to get closer and closer to the true value (μ)

Binomial Distributions

Binomial distributions can be used to show how probabilities can be used to assess the likelihood that an event will or will not occur given N observations.

Sometimes an event happening or not happening is referred to in terms of successes and failures.

Binomial Distribution

Coin tosses are a perfect example, because you can specify tossing heads or tossing heads as an event. Sticking with heads as the event, it either happens or fails to happen.

Critical Regions and ValuesIf we have established a critical region such that

we will reject the null hypothesis at 0, 1, 9 or 10 heads, then the size of the critical region would be calculated as follows:

p0 + p

1 + p

9 + p

10 = α (Critical region)

.001 + .01 + .01 + .001 = .022

So we have .022, or just a little more than 2 out of 100 chances of incorrectly rejecting the null hypothesis.

Critical Regions and Values

On a practical level, the only way one would reject the null hypothesis (H

0: P = .5) is if in 10 tosses

only 1,2,9 or 10 came up heads – none of which is likely with a fair coin.

Critical Regions and Values

In political science, the critical regions are typically referred to as levels. In other words, if α = .05 one would typically say “The null hypothesis can be rejected at the .05 level.”

This measure specifies the probability of making a Type I error (rejecting a true null hypothesis). This concept is also known as statistical significance.

Statistical Significance

The three most common levels of significance in political science are .05, .01 and .001.

Sometimes scholars use a looser standard of .10, .05 and .01.

Are these levels appropriate for the discipline?

One- or Two-Sided Tests

What if you suspect the null hypothesis is false? How would you go about formulating an alternative hypothesis?

Let's return to the coin tossing example. If I notice a coin tends to come up heads more likely than tails, I might propose H

A: P > .5 as an

alternative. This is different than merely assuming H

A: P ≠ .5 because prior observation

tells me there is a directional assumption that can be made. I'm not too worried that H

A: P < .5

One- or Two-Sided Tests

If testing a hypothesis theoretically suggests only upper or only lower values are relevant, a one-tail test will suffice.

In other words, a one-tail test requires only one critical region or value.

To return to the coin tossing example, if my HA:

P > .5 I will only be interested in the critical region where I get 9 or 10 heads out of 10 tosses, and therefore only interested in the critical value for the upper tail of the distribution.

One-Tail Test (High Values)

One-tail Test (Low Values)

One- or Two-Sided Tests

However, if I have to no reason to suspect large or small values of P, then I should use a two-tail test. .

In other words, if HA: P ≠ .5, I have no intuition

about whether the probability is higher or lower than .5, so I use a two-tail test.

What about real-world outcomes?

We obviously do not live in a binomial world. Usually we have to accept more than two

possible outcomes. As a result, a probability distribution would be increasingly difficult to tabulate.

Therefore, we cannot compare sample value to some critical value obtained from some distribution (such as the binomial distribution).

Types of Distributions

Discrete probability distribution

Continuous probability distribution

Discrete vs. Continuous Distributions (Kmenta 1986)

In discrete probability distributions the elements of sample space are represented by points that are separated by finite distances.

To each point we can ascribe a numerical value and to each value we can ascribe a given probability. (Ex: Coin toss (or binomial distribution), playing cards, lottery)

However, there are many distributions for which the sample space does not consist of countable points but covers and entire interval (or collection of intervals). These are known as continuous probability distributions.

Discrete Distribution

Continuous Distribution

Observed Test Statistic

Observed test statistic =

Sample estimate – hypothesized pop. parameterStandard error

The observed test statistic is compared to a critical value, and the decision to reject or not reject the null hypothesis depends on the outcome of the comparison.

Observed Test Statistic

1. If the observed statistic's value is greater than or equal to the critical value, reject the null hypothesis in favor of the alternative.

2. Otherwise, do not reject the null hypothesis.

Example of hypothesis testing

Example: Someone tells you “The average American has left the middle of the road and now tends to be somewhat conservative.” (H

0 : μ

= 5)

You, however, are not so sure. In light of Obama's recent election, you think America is not conservative. You believe it to be at least middle of the road. (H

A : μ < 5)

Example of hypothesis testing

Suppose you and your opponent decide to test these competing claims by examining mean voter ideology from the National Election Study (NES), which uses the following scale:

1 – Extremely liberal2 – Very liberal3 – Somewhat liberal4 – Moderate 5 – Somewhat conservative6 – Very conservative7 – Extremely conservative

Example of hypothesis testing

1 – Extremely liberal2 – Very liberal3 – Somewhat liberal4 – Moderate 5 – Somewhat conservative (opponent's claim)6 – Very conservative7 – Extremely conservative

H0 : μ = 5 – opponent's claim

HA : μ < 5 – your claim (μ is between 1 and 4)

Example of hypothesis testing

Before we start, we must decide on the size of the critical region. Let's set α = .05 (level of significance).

Next, we must specify the appropriate sampling distribution.

In a small sample (less than 25 observations) statistical theory asserts the appropriate sampling distribution fir a test about the mean is the t distribution.

The t distribution

The t distribution resembles a normal distribution but is a bit “fatter” in that it has more area in its tails.

The t distribution depends on the size of the sample (N). As N gets larger, the t distribution approaches the shape of the normal distribution; at N = 30 or N = 40 they are essentially indistinguishable.

In other words, use the t distribution if the sample is smaller than 30 or 40; use the normal distribution if N > 40.

t

Normal

To use a t distribution ...1. Determine the size of the sample to be

collected (rule of 30).

2. Find the degrees of freedom (df) by calculating N-1. Will explain df later.

3. Choose level of significance and directionality of the test, a one- or two-tailed test at the α level.

4. Given these choices, find the critical values located in Appendix B (for t distribution) in JRM pp. 576.

To use a t distribution ...

At this point you would now collect the sample data, find the sample mean and compute the observed test statistic (which in this case is a t-score).

The calculated t-score for the observations is then compared to a critical value t-score.

To use a t distribution ...

If the absolute value of the t-score for the observations is greater than the t-score for the critical values, reject H0. Otherwise, do not reject.

If |tobs

| ≥ tcrit

reject H0

If |tobs

| < tcrit

do not reject H0

To use a t distribution ...

is the sample mean is the hypothesized population mean is the sample standard deviation is the sample size

To use a t distribution ...

1. Sample size: N = 25.

2. Degrees of freedom (N-1) = 25 – 1 = 24.

3. One-tailed test; α = .05 (level of significance).

4. Look up the corresponding row for degrees of freedom and column for level of significance in

Appendix B for t distributions on page 576 to get the corresponding critical value.

To use a t distribution ...

Now calculate the t-score for the observations. In order to make the calculation we need the four following pieces of information: the sample mean, hypothesized population mean, sample standard deviation and sample size.

4.44 is the sample mean 5 is the hypothesized population mean 1.23 is the sample standard deviation 25 is the sample size

To use a t distribution ...

The observed t-score is -2.28. The critical value t-score is 1.711.

Again, if |tobs

| ≥ tcrit

reject H0

Since |-2.28| ≥ 1.711, H0 is rejected.

P-Values

The p-value tells you the probability of getting a t statistic at least as large as the one actually observed if the null hypothesis is true.

In this sample, the p-value is .016, which tells you the probability of getting a t statistic at least as large as the one actually observed if the null hypothesis is true.

In this sample there is only 1.6 percent chance of observing a as large as 4.44 if the population parameter is 5.

What about large samples?

Large samples rely on the standard or normal distribution, but how is the test statistic calculated?

The test statistic for a normal distribution is known as a z score, which is the number of standard deviations by which a score deviates from the mean score.

For example, z = 1.96 means 1.96 standard deviations above the mean.

How is a z-score calculated?

Z scores are calculated the the same way as t scores.

However, one has to use a different table to identify the appropriate critical value. The table for normal distributions (z scores) is Appendix A (JRM p. 575)

Example in Practice

Let's return to the ideology example. Let's assume we want to test the assumption that the United States has become a slightly conservative country according to the mean response in the NES (H

0 : μ = 5).

This time, however, you have no inclination about whether the null hypothesis is too conservative or too liberal. On the one hand, a fairly liberal presidential candidate just won the election, but on the other the United States has always been more conservative than most advanced industrial democracies (H

A : μ ≠ 5)

Example in Practice

Suppose we wanted a higher level of confidence. This time, we set the size of the critical region or probability to .01 (or α = .01)

Remember, the alternative hypothesis does not specify a relationship (that is, no less than or greater than).

Do we need a one- or two-tail test?

Example in Practice

Two-tail test. Whenever looking up the corresponding z score

for a critical region with a two-tail test, one has to divide the size of the critical region (here, .01) by 2.

So, .01/2 = .005. .005 is the size of the critical region in each tail. In total, the critical region is .01, giving us a 99

percent level of confidence. There is only a 1 % chance of committing a

Type I error.

To recap ....

If |zobs

| ≥ zcrit

reject H0

If |zobs

| < zcrit

do not reject H0

To use the z-score table ....

Notice how the values are arranged from largest to smallest, descending as you go across each row and continuing in descending order as you go down a row.

Find the value closest to .005 (Hint: It's .0049 on the table)

Add the number in the corresponding far left end of the row to the “second decimal place of Z” number at the top of the critical value's column.

For .0049, these numbers are 2.5 + .08, so 2.58 is the critical value of Z.

Since | -15.21 | > 2.58 we can reject the null hypothesis with 99 percent confidence.In other words, there's only a 1 % chance the null hypothesis is true. Put yet another way, the chance thatthe true population parameter for ideology being 5 is verysmall.

So what does this tell us?

The sample mean is 4.27, which is still in a slightly conservative direction.

In interpreting this statistic, a political scientist may conclude the United States is middle the road or perhaps slightly conservative, but not somewhat conservative (μ = 5).

Difference between t and z scores

t scores are used for samples of 30 or less; z scores for samples of more than 30.

t scores require us to calculate degrees of freedom; z scores do not require such a calculation.

The bigger the sample size, the smaller the size of the critical region for both t and z scores.

Example in Practice