Chapter 8 McGrewElements of Inferential Statistics

Dave MuenkelGeog 3000

Outline

• Classical Hypothesis Testing• P-Value Hypothesis Testing• One Sample Difference of Means Test• One Sample Difference of Proportions Test• Issues in Inferential Testing / Test Selection

Hypothesis

• A statistical hypothesis is simply a claim about a population that can be put to the test by drawing a random sample

Hypothesis Testing in Geography

• Make statements regarding unknown population parameter values based on sample data in order to:

- Refine Spatial Models - Develop Laws and Theories• A properly created sample is essential to Inferential

Statistics

Classical Hypothesis Test

• Steps:– State Null Hypothesis - Statement regarding the value of an unknown

parameter. Usually implies no association between explanatory and response variable.

– State Alternative Hypothesis - Statement contradictory to the null hypothesis.

– Select Test Statistic - Quantity based on sample data and null hypothesis used to test between null and alternative hypotheses

– Select Rejection Criteria – The value of the test statistic in which we reject the null or the alternative hypothesis

– Calculate the Test Statistic– Make a Decision regarding the Hypothesis

State the Hypothesis

The null hypothesis, Ho: Specifies hypothesized values for one or more of the population parameters

The alternative hypothesis, HA: A statement which says that the population parameter is something other than the value specified by the null hypothesis

Null and Alternative Hypothesis

The typical claim is that is equal to some value H (hypothesized mean). This claim of equality is called the Null Hypothesis.

Ho: 1 - 2 = 0, or Ho: 1 = 2

The Alternative Hypothesis is the alternate Hypothesis and expresses the condition for rejecting the Null Hypothesis.

HA: 1 - 2 0, or HA: 1 2

The two Hypotheses are mutually exclusive

Example Hypotheses

• H0: μ1 = μ2

• HA: μ1 ≠ μ2 – Two-sided test

• HA: μ1 > μ2

– One-sided test

Type I and Type II Error

State of the World Ho Accepted Ho Rejected

If Ho is true Correct decision Type I error

Pr = 1- Pr = If Ho is false Type II error Correct decision

Probability = Probability = 1 -

Select the Statistical Test(www.wikipedia.org)

Statistical Symbols(www.wikipedia.org)

Select Level of Significance

• If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%

Select Rejection Criteria

Calculate Test Statistic

0

/

XZ

S n

Test Statistic:

Make a Decision

The rejection of the null hypothesis implies the acceptance of the alternative hypothesis

Involves

Estimation

Hypothesis Testing

Purpose

To make decisions about population characteristics

Compare Test Statistic to Rejection Region

Upper-Tailed

Lower-Tailed

Two-Tailed

Make Decision on Hypothesis

fail to reject

rejectreject(1 a)

a/2a/2

P-value

• The smallest α the observed sample would reject H0

• If H0 is true, probability of obtaining a result as extreme or more extreme than the actual sample

• Is based on a modelNormal, t, binomial, etc.

Determining Statistical Significance: P-Value Method

• Compute the exact p-value (X.XX)• Compare to the predetermined α-level (0.05)• If p-value < predetermined α-level– Reject H0

– Results are statistically significant• If p-value > predetermined α-level– Do not reject H0

– Results are not statistically significant

Difference of Means / Proportions Test

• Used to compare a mean / proportion from a random sample to the mean of a population.

• Used to compare a mean / proportion from a random sample to the mean of a population.

• Assume Normal Distribution• For Large Samples use Z-Score• For small samples less than 30, use Students t

distribution

One sample difference of means z test

Degrees of Freedom

• the number of values in the final calculation of a statistic that are free to vary

• the minimal number of values which should be specified to determine all the data points

• whenever a parameter must be estimated to calculate a test statistic, a degree of freedom is lost

Inferential Test Selection

Consider - population of interest - investigative variables - sample data - inference about population based on sample data- reliability measure for the inference

Parametric and Non-parametric Tests

• Parametric tests – for particular assumptions about the underlying

population distributions– usually normal population is assumed

• Non-Parametric Tests– may be used on any distribution– with nominal ordinal data--only non-parametric

tests can be used