A.P. STATISTICS EXAM REVIEW TOPIC #2 Tests of Significance and Confidence Intervals for Means and Proportions Chapters

Vocabulary and key concepts Logic of significance testing Null and alternative hypotheses One- and two-sided tests P-values Statistically significant Type I and Type II errors Format of a significance test 2

Be sure you understand --- Significance tests begin with a claim about a population parameter (as stated in the null hypothesis). The alternative hypothesis can often be thought of as the research hypothesis. Significance tests are designed to determine whether or not an observed outcome is likely to have occurred if the null hypothesis is true. 3

The P-value of a test gives the probability that the observed outcome, or one even more different from the claim of the null hypothesis, would occur just by chance if the null hypothesis were true. 4

Small P-values indicate that there is little chance that the observed outcome, or one even more different from the claim of the null hypothesis, would occur just by chance if the null hypothesis were true. Small P-values lead us to reject the null hypothesis. Results with small P-values are said to be statistically significant; that is, the observed outcome is too different from the claim of the null hypothesis to attribute to chance. 5

Large P-values indicate that the observed outcome, or one even more different from the claim of the null hypothesis, is not rare and would be likely to occur just by chance if the null hypothesis were true. Large P-values lead us to conclude that the null hypothesis should not be rejected. Your conclusion should be either to reject the null hypothesis or fail to reject the null hypothesis. 6

NEVER conclude to... Accept the null hypothesis or Retain the null hypothesis ALWAYS conclude to either: Reject the null hypothesis or Fail to Reject the null hypothesis 7

How Does  Connect to the P-value? The significance level for a test is denoted by  The significance level tells us how small the P-value must be to reject the null. If the P-value is less than or equal to  the results are statistically significant at that  level. 8

Errors A Type I error occurs when you reject a null hypothesis that is true.  is the probability of a Type I error. A Type II error occurs when you fail to reject a null hypothesis that is not true.  is the probability of a Type II error 9

HCCC – a way to remember H - write hypotheses C - state and check conditions C – calculations C - state your conclusion in context 10

Defining Parameters Use words provided in the problem stem. Be sure to use parameter notation in your Hypotheses – Use  not x-bar, & p, not p-hat Include the population as part of the definition. 11

Writing Hypotheses ALWAYS write these using the PARAMETER!!!! I repeat: ALWAYS write these using the PARAMETER!!!! (  not x-bar, p not p-hat) H O is a statement of equality or status quo Ex. H o :  = 10 H A states the research hypothesis Ex. H A :  > 10 12

Conditions For each test or interval, be sure to list and verify the conditions (assumptions). There are different conditions for each type of interval and test. You need to memorize which set goes with each type. A list of these is in the packet. Don’t just check off the conditions with a check mark. Be sure to verify them in context, providing numbers and graphs where appropriate. 13

Name, Formula, Test Statistic (and if needed, Degrees of Freedom) In conjunction with verifying conditions, you need to name the test. Then you must calculate the test statistic. If the test statistic uses degrees of freedom, you need to state the df used. If you state the name of the test, statistic, and df, you will not be required to compute the test statistic by formula unless the problem specifically says to. 14

However... Showing the correct formula and/or numerical substitutions used to obtain the test statistic can earn you additional holistic credit! 15

P-value P-values can be obtained from the calculator. When you use the calculator statistical tests, you are provided with three choices for the alternative hypothesis. For tests for proportions, the plain “p” is the P-value. “Prop” is the population proportion and “ ” is the sample proportion. 16

Decision You will need to make a decision based on the P-value. Say... “Since the P-value =.012 <  =.05, we will reject the null hypothesis...” Or... “Since the P-value =.21 >  =.05, we fail to reject the null hypothesis... ” 17

Conclusions in Context Your conclusion should always be in the same pattern. For example, suppose you were testing: H O :  8 oz. vs H A :  8 oz. Sentence #1: Since the P-value =.014 is less than  =.05, we reject H O. Sentence #2: The data does lead me to believe that the population mean weight of a Snickers Bars is less than 8 oz. OR Since the P-value =.064 is higher than  =.05, we fail to reject H O. The data does not lead me to believe that the population mean weight of a Snickers Bars is less than 8 oz. 18

Which Distribution? For anything involving MEANS…. For anything involving PROPORTIONS… 19 T Z