9.1b h.w: pg 546: 9 – 13 odd

1. Tests with Fixed Significance LevelTarget Goal:I can reject or fail to reject the null hypothesis at different significant levels.I can determine how practical my results are. 9.1b h.w: pg 546: 9 – 13 odd

2. A level of significance α says how much evidence we require to reject Ho in terms of the P-value. • The outcome of a test is significant at level α if P ≤ α.

3. Ex: Determining Significance • In ex. “Can you balance your checkbook?” we examined whether the mean NAEP quantitative scores of young Americans is less than 275. • Ho: μ= 275, Ha: μ < 275 • The the z statistic isz = -1.45.

4. Is this evidence against Ho statistically significant at the 5% level? • We need to compare z with the 5%critical value z* = 1.645 from table A. • Why? Because z =-1.45 is not farther away from 0 than -1.645, it is not significant at level α = 0.05.

5. Ex: Is the Screen Tension OK? Recall proper screen tension was 275mV. Is there significant evidence at the 1% level that μ ≠ 275? Step 1:State -Identify the population parameter. • We want to assess the evidence against the claim that the mean tension in the population of all video terminals produced that day is 275 mV at 1% level. • H0 : μ = 275 • HA : μ ≠ 275 No change in the mean tension. There is change in the mean tension. (two sided)

6. Step 2: Plan Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. Since standard deviation is known, we will use a one-sample z test for a population mean. We checked the conditions before.

7. Step 3:Do - If the conditions are met, carry out the inference procedure. • Calculate the test statistic • Determine significance at the 1% level Because Ha is two sided, we compare = 3.26 with that of α/2 = .005 critical value from table C (two tails with total .01).

8. The critical value is z* = 2.576invNorm(.995) Step 4:Conclude -Interpret your results in the context of the problem. • Since z = 3.26 is at least as far as z* for α = 0.01, we reject the null hypothesis at the α = 0.01 sig. level and conclude that the screen tension is not the desired 275 level. 3.26

9. This does not tell us a lot. P-value • The P-value gives us a better sense of how strong the evidence is! • P-value = 2P(Z ≥ 3.26) = 2(normcdf(3.26,E99)), = 2(.000557) = .001114 • Knowing the P-value allows us to assess significance at any level. • We can estimate P-values w/out a calc (table A).

10. Test from Confidence Intervals The 99% confidence interval for the mean screen tension. • μ is = = (281.5, 331.1) Or, STAT:Tests:ZInterval:Stats (try!)

11. We are 99% confident that this interval captures the true population mean of all video screens produced. (281.5, 331.1) • Our value was 275. This does not fall in the range so • H0 : μ = 275 is implausible; thus we conclude μ is different than 275. • This is consistent with our previous conclusion.

12. Significance tests are widely used in reporting the results of research in many fields: • Pharmaceutical companies • Courts • Marketers • Medical Researchers Reading is fun!

13. Fixed Significance Levels • Chose α by asking how much evidence is required to reject Ho? • How plausible is Ho?If Ho represents an assumption people have believed for years, strong evidence (small α) will be needed. • What are the consequences for rejecting Ho? • If rejecting Ho in favor of Ha means an expensive changeover from one type of packaging to another, you need strong evidence the new packaging will boost sales.

14. 5% level (α = 0.05) is common but there is no sharp border between “significant” and “insignificant” only increasingly strong evidence as the P-value decreases. • There is no practical distinction between 0.049 and 0.051.

15. Statistical Significance and Practical Significance • Rejection of H0 at the α = 0.05 or α = 0.01 level is good evidence that an effect is present. • (But that effect could be very small.) Reading is fun!

16. Ex. 1 Wound Healing Time • Testing anti-bacterial cream: mean healing time of scab is 7.6 days with a standard deviation of 1.4 days. • Our claim is that formula NS will speed healing time. • We will use a 5% significance level.

17. Reading is fun! • Procedure: They cut 25 volunteer college students and apply formula NS. The sample mean healing time x = 7.1 days. We assume σ = 1.4 days.

18. Step 1: State We want to test claim about the mean healing time μ in the population of people treated with NS at the 5% significance level. • H0 :μ = 7.6 mean healing time of scabs is 7.6 days • Ha :μ < 7.6 NS decreases healing time of scabs

19. Step 2: Plan Since we assume σ = 1.4 days, use a one-sample z test. Random: The 25 subjects are volunteers so they are not a true SRS. We may not be able to generalize. Normal:Our sample is 25, proceed with caution. Independent: We can assume that the total number of college students is > 10(25).

20. Step 3:Do Compute the test statistic and find the p – value. Standardize: P( < 7.6) = P(Z < -1.79) = .0367

21. Step 4: Interpret your results in the context of the problem. • Since our p value, .0367 <α = 0.05 we reject Hoand conclude that NS healing effectis significant. Is this practical? • Having your scab fall off half a day sooner is no big deal. (7.6 days vs. 7.1 days) Reading is fun!