Tests with Fixed Significance Level Target 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

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

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 is z =

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

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. H 0 : μ = 275 H A : μ ≠ 275 (two sided) No change in the mean tension. There is change in the mean tension.

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.

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

The critical value is z* = invNorm(.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

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( ) = Knowing the P-value allows us to assess significance at any level. We can estimate P-values w/out a calc (table A).

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!)

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 H 0 : μ = 275 is implausible; thus we conclude μ is different than 275. This is consistent with our previous conclusion.

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

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

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 and

Statistical Significance and Practical Significance Rejection of H 0 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 !

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.

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. Reading is fun!

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. H 0 : μ = 7.6 mean healing time of scabs is 7.6 days H a : μ < 7.6 NS decreases healing time of scabs

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).

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

Step 4: Interpret your results in the context of the problem. Since our p value,.0367 < α = 0.05 we reject H o and conclude that NS healing effect is 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!