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If we live with a deep sense of gratitude, our life will be greatly embellished.

If we live with a deep sense of gratitude, our life will be greatly embellished. Hypothesis Test I: Z tests. Logic of hypothesis test Rejection region and p-value Test for one proportion Test for two proportions. Example: New Drug.

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If we live with a deep sense of gratitude, our life will be greatly embellished.

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  1. If we live with a deep sense of gratitude, our life will be greatly embellished.

  2. Hypothesis Test I: Z tests Logic of hypothesis test Rejection region and p-value Test for one proportion Test for two proportions

  3. Example: New Drug • A pharmaceutical company wants to be able to claim that for its newest medication the proportion of patients who experience side effects is less than 20%. Q. What are the two possible conclusions (hypotheses) here?

  4. The Logic of Hypothesis Test “Assume Ho is a possible truth until proven false” Analogical to “Presumed innocent until proven guilty” The logic of the US judicial system

  5. Steps in Hypothesis Test • Determine the null (Ho) and alternative (Ha) hypotheses • Find an appropriate test statistic and pre-set the level of significance (called a level) • Assuming Ho is true, find rejection region. • Reject Ho if the test statistic falls into the rejection region. • Report the result in the context of the situation Alternative steps for 3 & 4: 3. Assuming Ho is true, find p-value. 4. Reject Ho if p-value < a level.

  6. Determine Ho and Ha • Ho: nothing is happening (no relationship, no difference,…) • Ha: something is happening (there is a relationship, there is a difference, …) • Rule of Thumb: The “=“ sign must be in Ho. • If possible, set what we hope to prove as Ha.

  7. Example • p = % of users who will experience side effects Q. What are Ho and Ha here?

  8. Example Logic: Assume Ho is possibly true until proven false. Data: 17% of 400 patients who have experienced side effects How likely is if Ho is true (p>20%)? If very unlikely  reject Ho If not very unlikely  cannot reject Ho

  9. Rejection Region How extreme (i.e. unlikely) is the observation is too extreme? • Rejection Region is the region when the test statistic falls in, we will “reject Ho” • The rejection region is the most extreme region, determined by the a level and the type of Ha

  10. Two Types of Errors Type II Error, or “ Error” Good! (Correct!) we accept H0 Type I Error, or “ Error” Good! (Correct) we reject H0 H0 true H0 false

  11. Level of Significance • Level of significance also called a level is the largest tolerable type I error rate • Example of rejection region for a given a level

  12. P-Value • p-value is the smallest type I error rate if we reject Ho at the observed value • That is, p-value is the probability of seeing as extreme as (or more extreme) what we observe, given Ho is true. • The smaller p-value is, the less likely that what we observe will occur given Ho is true. • That is, smaller p-value means stronger evidence against Ho.

  13. P-Value • The level of significance (called a level) is usually 0.05 • p-value > a fail to reject Ho (??) • p-value <a reject Ho (= accept Ha)

  14. Report the Conclusion • Reject Ho: the data shows strong evidence supporting Ha Eg. The data shows strong evidence that the proportion of users who will experience side effects is less than 20%. • Fail to reject Ho: the data does not provide sufficient evidence supporting Ha Eg. Based on the data, there is not sufficient evidence to support the proportion is less than 20%

  15. Testing Hypotheses about a Proportion • Three possible Ho and Ha Write them all as p=po in the future

  16. The z-test for a Proportion • When 1) the sample is a random sample 2) n(po) and n(1-po) are both at least 5, an appropriate test statistic for p is

  17. Computing the p-Value for the Z-Test

  18. Computing the p-Value for the Z-Test

  19. Computing the p-Value for the Z-Test P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)

  20. Example: New Drug (Conti.) • Ho: p > 20% vs. Ha: p < 20% • Z-test statistic; a = 0.05 • Find rejection region or p-value • Decide if reject Ho or not • Report the conclusion in the context of the situation

  21. Hypothesis Test for the Difference of Two Population Proportions • Step 1. Set up hypotheses Ho: p1 = p2 and three possible Ha’s: Ha: p1 = p2 (two-tailed) or Ha: p1 < p2 (lower-tailed) or Ha: p1 > p2 (upper-tailed)

  22. Hypothesis Test for the Difference between Two Population Proportions • Step 2. calculate test statistic where

  23. Hypothesis Test for the Difference between Two Population Proportions • Step 3: Find p value • Must be two independent random samples; both are large samples: And 2. When the above conditions are met, use Z-Table to find p-value.

  24. Example: Bike to School For 80 randomly selected men, 30 regularly bicycled to campus; while for 100 randomly selected women, 20 regularly bicycled to campus. • Find the p-value for testing: Ho: p1 = p2 vs. Ha: p1 > p2 Answer: z=2.60, p=0.0047 1: men; 2: women

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