WARM – UP Are twin births on the rise in the United States? In % of all births produced twins. In 2009 Data from an SRS of 469 mothers in a large city found that 4.05% resulted in sets of twins being born. Test the appropriate hypothesis and state your conclusions and assumptions. p = The true proportion of twins births in the United States. H 0 : p = 0.03 H a : p > 0.03 Since the P-Value is NOT less than α = 0.05 we Fail to REJECT H 0. There is no evidence to suggest that Twin births are on the rise in the US. 1.SRS – The data was collected randomly 2.Appr. Normal: 469 · (0.03) ≥ 10 AND 469 · (1 – 0.03) ≥ 10 3.Population of US births is ≥ 10 · (469) One Proportion z – Test

H 0 = NO CHANGE in Population, No effect, THE NORM, or THE EXPECTATION. ONE-Tailed Test – H a : p > #orH a : p < # TWO-Tailed Test : H a : p ≠ #

Chapter 21 STATISTICAL ERRORS If we Reject H 0 when in fact H 0 is TRUE this is a TYPE I ERROR = (false positive). (α = probability of committing a Type I Error.) If we Fail to Reject H 0 when in fact H 0 is FALSE (H a is True) this is a TYPE II ERROR = (false negative). (β = probability of committing a Type II Error.) Decision based on sample Reject H 0 Fail to Reject H 0 H 0 is True H 0 is False Type I Error Correct Decision Type II Error

In each example List the Hypothesis and then Describe the Type I and Type II Error. Page In 2003 the Dept. of Commerce reported that 68.2% of American families own homes. In order to encourage home ownership the city offers a plan of tax breaks for first time home buyers. Since the plan cost the city tax revenues, they will continue using it only if there is strong evidence that the rate of home ownership is increasing. H 0 : Home ownership remains Unchanged Not rising. H a : Home ownership is Increasing. TYPE I ERROR = The city feels that home ownership is on the rise but in fact the tax breaks are not helping. The city forgoes tax revenues for nothing. TYPE II ERROR = The city feels homeownership remains unchanged (68.2%), but it is increasing. They may retract the tax breaks feeling they are useless.

In each example List the Hypothesis and then Describe the Type I and Type II Error. 12. Recently a group of doctors devised a quick test to test for the Alzheimer in the population of senior citizens. A patient that tested positive would then go through a more expensive and time consuming battery of tests and medical diagnosis. H 0 : The Patient does not have Alzheimer (Healthy) H a : The Patient has Alzheimer TYPE I ERROR = The Patient is diagnosed with Alzheimer but does NOT have the disease. The Patient waste time and money… and stress. FALSE POSITIVE TYPE II ERROR = The patient is diagnosed as healthy when they do have Alzheimer. The patient does not receive the beneficial treatment. FALSE NEGATIVE

Wednesday’s Quiz 1. Definition P-Value: 2. Definition of Confidence Level: 3. Definition of Central Limit Theorem: 4. Proportion Assumptions/Conditions: 5. Decision/Conclusions based on p-value: 6. Definition of Type I and II Errors: 7. Construct and Interpret Confidence Interval 8. Perform Significance Test. Probability of obtaining statistics or ones more extreme, given H 0 is true. In Repeating Sampling C% of the constructed intervals will contain the True Parameter. A large random sample will produce an approximately normal distribution.

Homework- Page 492: 12a-d, 13

In each example List the Hypothesis and then Describe the Type I and Type II Error. 13. Clean Air standards require that vehicle exhausts not exceed specified limits. Many states require annual test. State Regulators sample shops all the time. They will revoke the shop’s license if they find that they are certifying vehicles that do not meet the standards. H 0 : The Shop is certifying cars that are meeting the standards. H a : The Shop is certifying cars that DO NOT meet the standards. TYPE I ERROR = The state regulator decides that the shop is not meeting the standards when they really are. The shop is fined and license revoked. TYPE II ERROR = The state regulators decide that the shop is following standards when it is NOT. Consequently the air is being polluted.

Power – The Probability that a significance test will correctly reject a false H 0. “Doing the right thing.” Power = 1 – β (Type II Prob.) Increasing sample sizes decrease Type II errors and consequently increase the Power of the test Decision based on sample Reject H 0 Fail to Reject H 0 H 0 is True H 0 is False Type I Error Correct Decision POWER Type II Error Correct Decision

Chapter 21 – More about Tests Common alpha levels are 0.05, 0.10, and – When we reject the null hypothesis, we say that the test is “significant at that level.” significance level.The alpha level, α =0.05,is also called the significance level.

Back to the Basics… 469 Are twin births on the rise in the United States? In % of all births produced twins. In 2005 Data from an SRS of 469 mothers in a large city found that 19 sets of twins were born to 469 mothers. Test the appropriate hypothesis and state your conclusions and assumptions. Because our sample size is 469 we get a Sampling Distribution of… N(0.03, ) Definintion of P-value: The Probability of obtaining this statistic, p = 19/469 = , given that the Null hypothesis of μ = 0.03 is TRUE. P-value =

Back to the Basics… What happens if you increase the sample size? Lets say that we obtain a sample of 2000 births of which 81 result in twins. Because our sample size is 2000 we get a Sampling Distribution of… N(0.03, ) Definintion of P-value: The Probability of obtaining this statistic, p = 81/2000 = , given that the Null hypothesis of μ = 0.03 is TRUE. P-value =

Decreasing α means that you would need more evidence to Reject Ho. But α has an INVERSE relationship with β. This means that if you increase α then you decrease β. Since Power is 1 – β then an increase in α makes for an increase in Power (Failing to reject a TRUE hypothesis.)