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Testing Hypotheses about Proportions

Testing Hypotheses about Proportions. Hypothesis tests, null hypothesis, P-values, one and two sided z-tests on a proportion. Comparing Hypothesis testing to a criminal trial 1- Put together a case. 2- Gather evidence . 3- Examine evidence. 4- State the verdict.

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Testing Hypotheses about Proportions

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  1. Testing Hypotheses about Proportions Hypothesis tests, null hypothesis, P-values, one and two sided z-tests on a proportion

  2. Comparing Hypothesis testing to a criminal trial 1- Put together a case. 2- Gather evidence . 3- Examine evidence. 4- State the verdict. We never go to trial to prove someone is innocent, we go to see if they are guilty so if we can’t declare them guilty, we declare them not guilty (never innocent, the trial wasn’t about innocence, it was about guilt). Ho: the status quo, Ha: what do you think the status quo is based on the sample? Are the conditions of our sample such that we can use the normal model? Find n, p, q, p-hat, , z-score, and the p-value Compare the p-value to  Reject the null or fail to reject the null – never accept it

  3. Comparing Hypothesis testing to a criminal trial 1- Assume the status quo {answer question 1} called a Null Hypothesis or H0 (assume that nothing has changed) *written as H0 : p = percentage as decimal* {answer question 2} What is the probability that the sample data could fall in the normal model based on the status quo? What difference does sample size make anyway?

  4. N( , ) Sample size n % chance it will happen in the model sample data status quo Confidence interval

  5. N( , ) Sample size 6n % chance it will happen in the model sample data status quo Confidence interval

  6. Comparing Hypothesis testing to a criminal trial • 1- Assume the status quo • {answer question 1} • called a Null Hypothesis or H0 (assume that nothing has changed) • *written as H0 : p = percentage as decimal* • {answer question 2} • So what is the alternative– what you think the status quo actually is. • called Alternative Hypoth. HA • 3 choices: HA : p ≠ percentage as decimal – just different • HA : p < percentage as decimal – less than status quo • HA : p > percentage as decimal – greater than status quo • {answer question 3} • {answer question 4}

  7. N( , ) HA : p > percentage as decimal HA : p < percentage as decimal status quo

  8. N( , ) HA : p ≠ percentage as decimal status quo

  9. Comparing Hypothesis testing to a criminal trial • 1- H0 and HA (we have suspicion of an incorrect p) • Can we go to trial? • Can we create a case that p is incorrect? (is it appropriate to go to trial)? • {answer question 5} • Independence Assumption • Randomization condition - SRS or representative sample • Sample Size; 10% condition (sample less than 10% of population) • success / failure (np and nq > 10) • If the case passes these, than its ok to use the normal model for p-hat • {answer questions 6-10} • Now its time to gather the data from your sample!! • Don’t eat anything until the data have been collected.

  10. Comparing Hypothesis testing to a criminal trial • 1- H0 and HA • 2- Can we go to trial? • 3- Gather the evidence: • need to know sample size n • need to know mean or p and q • need to know SD(p-hat) = • draw normal model to find  • ( is the point of rejection for the H0) • depends on 1 or 2 tails, • based on your confidence percent level (99, 95, 90 etc) • {answer question 11}

  11. N( , ) HA : p < percentage as decimal 95% 5%  = -0.05 status quo • {answer question 12}

  12. N( , ) HA : p ≠ percentage as decimal 95% 2.5% 2.5%  = -0.025  = 0.025 status quo

  13. Comparing Hypothesis testing to a criminal trial 1- H0 and HA 2- Can we go to trial? 3- Gather the evidence: (cont…) need to know z-score need to know P-value (percent of area beyond p-hat) {answer question 13}

  14. N( , ) HA : p < percentage as decimal Where does the z-score fall on the model? What is the % under the curve? 95% 5%  = -0.05 status quo

  15. Comparing Hypothesis testing to a criminal trial 1- Put together a case. 2- Gather evidence . 3- Examine evidence. compare your p-value to the value 4- Render Verdict a) State the p-value and its probability based on model b) reject the null if p-value is outside  (guilty) the status quo is rejected – there is a difference c) fail to reject null if p-value is within  (closer to p than then  is) (not guilty) everything remains status quo d) put it in context

  16. A report on health care in the US said that 28% of Americans have experienced times when they haven't been able to afford medical care. A news organization randomly sampled 801 black Americans, of whom 32% reported that there had been times in the last year when they had not been able to afford medical care. Does this indicate that this problem is more severe among black Americans? Run a hypothesis test and state your conclusion.

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