PARAMETRIC STATISTICAL INFERENCE

PARAMETRIC STATISTICAL INFERENCE

FIXED SIGNIFICANCE LEVEL

·       Final step in assessing evidence against Ho.

Sometimes we demand a specific degree of evidence in order to reject the null hypothesis

Level of significance - - specifies how

much evidence is required before we reject Ho

·       Compare p-value with a fixed value that we regard decisive

The outcome of a test is significant at level

if p-value

otherwise when p-value > , we retain Ho.

·       Significance level is chosen before experimentation

PARAMETRIC STATISTICAL INFERENCE

Example: If we choose = 0.05, we are requiring that the data give evidence against Ho so strong that the sample outcome would happen no less often than 5% of the time when Ho is true

Typical values of alpha lie between 0.01 and 0.10.

·       The smaller the level the stronger the evidence needs to be to reject Ho.

·       After p-value is calculated, it is compared to the chosen level

o      If p-value is as small or smaller than , we say that the data are STATISTICALLY SIGNIFICANT AT LEVEL

·       Significant = not likely to happen by chance 

PARAMETRIC STATISTICAL INFERENCE

Exercise:

1. Considering the previous example, would you accept or reject Ho for the following levels of alpha?

a. 0.05

b. 0.01

c. 0.001

2. Would you reject or accept Ho under the following circumstances for each of these one-sided hypotheses?

a. P=0.12; = 0.01

b. P=0.05; = 0.06

PARAMETRIC STATISTICAL INFERENCE

CRITICAL VALUES OF Z – z*

• The number z* with probability p lying to its right under the standard normal curve is called the upper p critical value of the standard normal distribution.

PARAMETRIC STATISTICAL INFERENCE

 TESTS FROM CONFIDENCE INTERVALS

• A level two-sided significance test rejects a hypothesis Ho: = o exactly when the value o falls outside a level 1- confidence interval for

• Example:The mean household size in a certain city

is 3.2 persons with a standard deviation of =1.6. A

firm interested in estimating weekly household expenditures on food takes a random sample of n=100 households. Calculate the 99%

confidence interval for . (a) Where does the sample score

of 3.6 households fall within this interval? (b) Did you

retain Ho for the 2-tailed alternative in your homework? Compare (a) and (b)

PARAMETRIC STATISTICAL INFERENCE

 MAKING SENSE OF STATISTICAL SIGNIFICANCE

• Significance tests are widely used in reporting results of research

• Statistical significance is valued because it points to an effect that is unlikely to occur simply by chance

• Carrying out tests is simple, using tests wisely is not so simple

HOW SMALL A P IS CONVINCING?

• Purpose of test of significance is to describe the degree of evidence provided by the sample against the null hypothesis – p-value does this

• But, how small a p-value is convincing evidence?• Answer depends on two circumstances:

– How plausible is Ho?• If Ho represents an assumption that the people you must convince

have believed for years, strong evidence will be needed to persuade them

– What are the consequences of rejecting Ho?• If rejecting Ho in favor of Ha means making an expensive changeover

from one type of belief to another you need strong evidence that the new belief should be adapted

• As general rule, always report your p-values and let the reader decide for him/herself if the results are significant.

• Remember: There is no sharp border between “significant” and “insignificant”, only increasing strong evidence as P-value decreases. Eg. P-value = 0.049 vs p-value = 0.051.

PARAMETRIC STATISTICAL INFERENCE

 STATISTICAL SIGNIFICANCE AND PRACTICAL SIGNIFICANCE

• When an null hypothesis can be rejected at the usual levels of 0.05 and 0.01, there is good evidence that an effect is present

• But, is this effect large enough from practical sense?

• When large samples are available, even tiny deviations from the null hypothesis will be significant

Example: We are testing the hypothesis of no correlation between 2 variables. With 1000 observations, an observed correlation of only r = 0.08 is significant evidence at the 0.01 level that the correlation in the population is not zero but positive.

• The low significance level does not mean there is a strong association, only that there is strong evidence for some association.

• We may well conclude that for practical purposes, we can ignore the association between these variables, even though the association is statistically significant

PARAMETRIC STATISTICAL INFERENCE

PARAMETRIC STATISTICAL INFERENCE

 VALIDITY OF STATISTICAL SIGNIFICANCE

• Statistical inference is not valid for all sets of data

• Badly designed experiments produce useless results

• A statistical test is only valid under certain circumstances

• Data must be properly produced – SRS

• Z-test should bear the same warning label that we attached to the confidence intervals