Math 113: Chapters 7-8 Study Guide

Describe in every day language what a Type I or Type II error is.

A Type I Error is to say false when true. (Everyday)
A Type II Error is to say true when false. (Everyday)
A Type I Error is to reject a true null hypothesis. (Statisically)
A Type II Error is to fail to reject a false null hypothesis. (Statisically)

Know which distribution should be used to test (4 questions) ...

Test	Distribution	Notes
Mean(s)	Z or T	Use Z if sigma is known Use T if sigma is unknown Two samples with Dependent case creates a new variable for which you then find the sample standard deviation, so use T.
Proportion(s)	Z	np >= 5, nq >= 5
Variance	Chi-Square
Variance(s)	F	Put the larger variance on top will make it be a right tail test.

Know the default level of significance.

0.05

Know the definition of the level of significance.

The probability of being in the critical region
The probability of being more extreme than the critical value
The probability of committing a Type I Error
The probability of rejecting a true null hypothesis

Know the definition of the critical value.

The value which separates the critical region from the non-critical region
The value which separates the values which cause rejection from the values which would not cause rejection

Know the definition of null and alternative hypotheses.

Null: Statement of no change
Null: Always contains the equal sign
Alternative: Statement of change
Alternative: Does not contain the equal sign

Know the definition of the probability-value.

The probability of being more extreme than the test statistic

Know the definition of the F-variable.

The ratio of two independent chi-square variables divided by their respective degrees of freedom.

Know the relationship between prob-value, test statistic, level of significance, and critical values.

The test statistic is to the critical value as the p-value is to the level of significance
The test statistic is to the p-value as the critical value is to the level of significance

Know the difference between the decision and conclusion as to whether the null hypothesis or original claim is used.

The decision is based on the null hypothesis.
The conclusion is based on the original claim.

Know the difference between critical value and test statistic as far as which is looked up and which is calculated.

The critical value is looked up
The test statistic is calculated

Know the results from the classical approach and how they differ from the results with the prob-value approach.

The classical approach and the probability-value approach will always have the same results.
P-value: Reject if the p-value is less than the level of significance, no matter what type of test it was.
Classical: Depends on the type of test
The p-value is easier for a computer to calculate
You can decide your own level of significance if you're given a p-value without having to look up another critical value
The end-user can look at a p-value and make a decision without having to know where the p-value came from

Know properties of the Standard Normal distribution.

Symmetric about its mean
Mean is 0
Standard deviation and variance are 1
About 68% lies within one standard deviation of the mean.
About 95% lies within two standard deviations of the mean.
About 99.7% lies within three standard deviations of the mean.
It's better suited to the p-value approach

Know properties of the Student's T distribution

Symmetric about its mean
Mean is 0
Standard deviation and variance are greater than 1
Requires degrees of freedom
Actually many distributions
Approaches the normal distribution as the sample size gets larger
Discovered by Irish Brewery worker William T. Gosset
Better suited to the classical approach

Know properties of the Chi-Square distribution.

Not symmetric
No negative values
Requires degrees of freedom
Actually many distributions
To look up a critical value on the left, you must first subtract the area on the left from one and then look it up.
Best suited to classical approach
Mean is its degrees of freedom
Variance is its degrees of freedom

Know properties of the F Distribution

Not symmetric
No negative values
Requires two different degrees of freedom, one for the numerator and one for the denominator
It is the ratio of two independent chi-square variables divided by their respective degrees of freedom
Mean is approximately 1
Placing the larger variance on top will make it a right tailed test
Requires several different tables, one for each level of significance
Best suited to the classical approach

Know which distributions are best used with the classical approach and which are best used with the prob-value approach.

P-value approach is best suited when the critical values are given on the outside of the table and the probabilities are looked up on the inside
Classical approach is best suited when the probabilities are given on the outside of the table and the critical values are looked up on the inside.
P-value approach works best with a Normal distribution.
Classical approach works best with a Student's t, Chi-Square, or F distribution

Know when to reject the null hypothesis using both the classical and prob-value approaches.

Prob-value: Reject if the prob-value is less than the level of significance
Classical: Reject if the test statistic is less than the critical value for a left tail test
Classical: Reject if the test statistic is greater than the critical value for a right tail test
Classical: Reject if the test statistic is less than the left critical value or greater than the right critical value for a two tail test
Classical: Reject if the test statistic is more extreme than the critical value

Look up the critical values for a test about a single population mean.

If you know sigma, then use Z
If you don't know sigma, then use T and let degrees of freedom = n-1
Use the appropriate row (one-tail or two-tail)
Make critical value negative for a left tail test
Give two critical values (positive and negative) for a two tail test

Look up the critical values for a test about a single population mean.

See previous problem.

Look up the critical values for a test about a single population variance.

Use chi-square table
Degrees of freedom is n-1
If right tail, then just look up critical value in table
If left tail, subtract level of significance from 1, and then look up.
If two tail, then divide the level of significance by two.

Look up that area for the right critical value
Subtract that area from one and look up for the left critical value

Do NOT make negative - remember, chi-squares are non-negative

Look up the critical values for a test about two population variances.

Use the F-table
Divide the level of significance by two for a two tail test
Place the larger variance in the numerator to make it a right tail test

Find the prob-value for a test statistic which isn't a value directly in the table. The best that you will be able to do is to say the prob-value is between two probabilities

Go to the appropriate degrees of freedom in the table
Find the two critical values on either side of your test statistic
Go to the top of the table and read the two probabilities for those columns.
The p-value will be between those two probabilities
Use the one-tail or two-tail rows depending on the type of test.
Do NOT make negative for a left tail test - these are probabilities, they can't be negative

Look up the critical value in a chi-square table when the degrees of freedom aren't in the table. You should go to the value in the table which is less likely to reject in error.

Use the critical value (inside of the table) which is more extreme

Larger critical value for a right tail test
Smaller critical value for a left tail test

Do not pick the larger or smaller degrees of freedom, pick the larger or smaller critical value.

Tell how you can tell whether the test for a single population mean is a left-tail, right-tail, or two-tail test by looking at the critical value(s).

Negative critical value implies left tail
Positive critical value implies right tail
Negative and positive critical values imply two tails

Calculate the test statistic used to test the equality of two population variances (F-test).

F is the ratio of the variances
Put the larger variance on top to make it a right tail test
Be sure to square if the standard deviations are given

A claim and level of significance are given. You are told the decision and you need to write the conclusion.

Look at the decision

Reject the null hypothesis means there is sufficient evidence
Fail to reject the null hypothesis means there is insufficient evidence

Determine if the original claim is the null or alternative hypothesis

If the original is the null hypothesis, then there is _____ evidence to reject the claim
If the original is the alternative hypothesis, then there is _____ evidence to support the claim.

Notes:

Problems 1 - 6 are short answer.
Problems 7 - 15 are true false.
Problems 16 - 21 are multiple choice
None of the problems are directly from the text (that I know of).