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3. Two Possible Results: page 85 When p-value ≤ , we Reject H 0 and say the results are statistically significant at level  When p-value > , we Fail to Reject H 0 and say the results are not statistically significant at level  Always … write a real-world conclusion in sentence form in context of problem.

4. Aside: Pick  before looking at data! pg 88 If = 0.08, certainly NOT reject H 0 If = 0.12, z=0.82, p-value=0.206  fail to reject H 0 If = 0.16, z=2.45, p-value=0.007  reject H 0 If = 0.19, z=3.7, p-value=0.0000  reject H 0 If = 0.29, z=7.8, p-value nearly 0  reject H 0

5. Try It! Left-Handed Artists page 87 About 10% of human population is left-handed. Researcher speculates artists more likely to be left-handed. Sample of 150 artists  18 are left-handed. Perform test at 5% significance level. Step 1: Determine hypotheses H 0 :H a : where the parameter ________ represents … Direction:

6. Try It! Households without Children US Census: 48% of households have no children. Random sample of 500 households taken to assess if population proportion changed from Census value of 0.48. Result: 220 households had no children. Use 10% sig level. Step 1: Determine hypotheses H 0 : p = 0.48 H a : p ≠ 0.48 where the parameter p represents the population proportion of all households today that have no children. Direction: TWO-SIDED

7. Try It! Households without Children Step 2: Verify conditions, summarize via test stat. Data assumed to be a random sample Check np 0  10 and n(1 – p 0 )  10 Test statistic:

8. Try It! Households without children Step 3: Find p-value. two-sided test => both large and small values are “extreme”. Sketch p-value: Compute p-value: Step 4 = Click in your answer …

9. Step 4: Decide if result is statistically significant. p-value = ____________ Statistically significant at the 10% level? Click in your answer: Yes or No

10. Try It! Households without children Step 5: Report conclusion in context of situation.

11. What if n is small? Page 89 Goal: Learn about a population proportion p Take a r.s. of size n. If sample size is small, go back to exact distribution for count X  binomial distribution. If X has the binomial distribution Bin(n,p), then where for k = 1, 2, …,n

12. Small Sample Binomial Test To test hypothesis H 0 : p = p 0 we compute the count test statistic X = number successes in sample of size n which has Bin(n, p 0 ) distrib when H 0 true This Bin(n, p 0 ) distribution is used to compute the p-value for the test.

13. Try It! New Treatment page 90 10 subjects treated with a new treatment, 9 improved. Test claim “a majority” of people using treatment improve using a 5% significance level. Let p = popul proportion of people who improve with trt. H 0 : H a :

14. Try It! New Treatment 10 subjects treated with a new treatment, 9 improved. Observed test statistic value is just: p-value =

15. Try It! New Treatment p-value = 0.0107 At the 5% significance level, we would _____________ and conclude: Get ready for a clicker question …

16. Besides being a small sample, which is another major issue regarding this study? A) Non-response bias B) Response bias C) It’s just an observational study

17. 12.5 Sample Size, Statistical Significance and Practical Importance page 91 From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

18. 12.5 Sample Size, Statistical Significance and Practical Importance The phrase statistically significant only means that the data are strong enough to reject the null hypothesis. The p-value tells us about the statistical significance of the effect, but it does not tell us about the size of the effect.

19. What can go wrong? Page 92 Type 1 error = rejecting H 0 when H 0 is true Type 2 error = failing to reject H 0 when H a is true P(Type 1 error) = P(Type 2 error) = Power of the test = P(reject H 0 when H a true) = 1 – P(fail to reject H 0 when H a true) = 1 – P(Type 2 error) = 1 – 

20. Comments: 1.In practice want to protect status quo so most concerned with 2.Most tests described have the 3.Generally, for fixed sample size n, 4.Ideally want probabilities of making a mistake to be small. However,…

21. Comments: 5.Three factors that influence power  Sample size: larger sample size  higher power  Significance level: larger   higher power  Actual parameter value: true value further from null value (in direction of alternative hyp)  higher power   From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

22. Simple Example: page 93 H 0 : Basket has 9 Red and 1 White H a : Basket has 4 Red and 6 White Data: 1 ball selected at random. Reasonable DR: Reject H 0 if the ball is ___________ P(Type 1 error) = P(Type 2 error) = Power =

23. Simple Example: H 0 : Basket has 9 Red and 1 White H a : Basket has 4 Red and 6 White DR: Reject H 0 if the ball is White Suppose ball selected and observed to be WHITE. What would be the decision? ________________ Could a mistake have been made? If so, which type? What is probability this type of mistake was made?

24. 2 nd page of Yellow Card on the Big 5 Parameters We have covered the first column, moving into column 2