THE BIG PICTURE HANDOUT 9.1 PART ONE: General Information about Hypothesis Testing Writing the claim, Ho, and H1. -Identify the claim stated in the problem and express it in symbolic form. -Give the symbolic form of the claim that must be false if the claim is true. -Of the two symbolic expressions obtained so far, let the alternative hypothesis H1 be the one not containing the equality so H1 uses the symbol or < or >. -Let the null hypothesis Ho be the symbolic expression that the parameter equals the fixed value being considered. Example) Test the claim that the mean is greater than 4. Claim: >4 BUT 4 will be false when the claim is true so we choose the statement with or < or > to be H1 and Ho always has the = sign.) Ho: =4 H1: >4 Example) Test the claim that the mean is equal to 10. Claim: =10 BUT 10 will be false when the claim is true so we choose the statement with or < or > to be H1 and Ho always has the = sign.) Ho: =10 H1: 10 Example) Test the claim that the mean is less than or equal to 5. Claim: 5 BUT >5 will be false when the claim is true so we choose the statement with or < or > to be H1 and Ho always has the = sign.) Ho: =5 H1: > 5 Example) Test the claim that the mean is not equal to 7. Claim: 7 BUT =7 will be false when the claim is true so we choose the statement with or < or > to be H1 and Ho always has the = sign.) Ho: =7 H1: 7
µ µ ≤ ≠
µ µ ≠ ≠
µ µ ≠
µ ≤ µ ≠
µ ≠ µ ≠
µ µ ≠
Symbols Mean Standard
Deviation Variance Proportion Correlation
e Population Parameters
“beta sub 1”
s “P hat”
Note: The population parameters are used in the claim, Ho, and H1. Rules for the p value method: for DECISION about Ho If , then reject Ho.
If , then we do not reject Ho. Note: We assume Ho is true from the start. If based on the sample data, Ho is unusual, then we reject Ho. “Unusual” is less than or equal to the alpha level. For example) = .05 or 5%. In this case we assume Ho is true unless the probability of Ho occurring based on the sample data, p, is unusual which is 5% or less. Case 1: = .05 or 5% and p = .03 or 3% so , so we reject Ho.
Case 2: = .05 or 5% and p = .10 or 10% so , then we do not reject Ho. To write the LONG CONCLUSION about the CLAIM Decision Claim Claim has “condition of
equality” means Claim has or =
Claim does not have “condition of equality” means Claim has or < or >
Reject Ho There is enough evidence to reject the claim.
There is enough evidence to support the claim.
Do not Reject Ho There is not enough evidence to reject the claim.
There is not enough evidence to support the claim.
Hypothesis testing is like the courtroom…. we always assume innocence unless proven guilty. We always assume Ho is true unless there is enough evidence to show otherwise. That is why when the claim has the condition of equality which is similar to Ho, we use the word REJECT and when the claim does not have the condition of equality which is unlike Ho, we use the word SUPPORT.
µ σ σ 2 ρ β1
x s 2
p ≤ α p > α
α p ≤ α α p > α
≥ or ≤ ≠
CONCLUSION about Ho or H1 If there is sufficient evidence to reject Ho, then we conclude that H1 is true. If there is not enough evidence to reject Ho, then we conclude that Ho might be true, but we never conclude that Ho is true. Recall in the courtroom, the verdict is man is guilty or man is not guilty, but never man is innocent. Note: When we do not reject Ho, it just means that the evidence was not strong enough to reject Ho. Interpreting Confidence Intervals Ex) A 95% confidence interval with n = 200, for the difference between two means, has already been calculated to be: (.381, .497). Correct Interpretation “We are 95% confident that the true value of lies in the confidence interval.” This means if we were to select many different samples (for example 100 samples) of size 200 and construct the corresponding confidence intervals, 95% of them (95) would actually contain the true value of and 5% (5) of them would not. Note: In this correct interpretation, the level 95% refers to the success rate of the process being used to estimate , and it does not refer to the difference of the population means. Incorrect Interpretation There is a 95% chance that the true value of lies in the confidence interval. Note: Where does come from? It comes from rewriting Ho. Ho:
µ1 − µ2
µ1 − µ2
µ1 − µ2
µ1 − µ2
µ1 − µ2
µ1 − µ2
µ1 = µ2 or µ1 − µ2 = 0
PART TWO: One Sample & Two Sample Tests Conditions How to know when to use which test? I. ONE-SAMPLE TESTS: MEAN Z test for a Mean is known and the population is normal OR for any population when
is known and n 30. T test for a Mean is not known and the population is normal OR for any population when is not known and n 30. PROPORTION 1-Prop Z test II. TWO-SAMPLE TESTS TWO MEANS 2 Sample Z-test Independent Samples. Both populations normal OR both sample sizes are at least 30. known. 2 Sample T-test Independent Samples. Both populations normal OR both sample sizes are at least 30. not known. T-Test for dependent samples Dependent samples. Both populations normal OR both sample sizes are at least 30. TWO PROPORTIONS 2-Prop Z Test TWO STANDARD DEVIATIONS 2 Sample F Test Table below summarizes parametric and nonparametric tests (do not require any specific conditions concerning the shapes of population distributions or the values of population parameters) Always use the parametric test if the conditions are satisfied.
σ σ ≥
σ σ ≥
σ1 and σ 2
σ1 and σ 2
TEST APPLICATION Parametric test ch 9-14
Nonparametric test Ch15
One -sample tests Z test for Mean Sign test for Median T test for Mean Two-sample tests Dependent T test Sign test/Wilcoxon
Signed Rank Independent 2 sample z test Wilcoxon Rank Sum 2 sample t test 3 or more samples One-Way ANOVA Kruskal- Wallis Test Correlation Pearson Correlation
Coefficient Spearman rank correlation coefficient
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