## 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.
• 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
• 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).