9.1b h.w: pg 546: 9 â€“ 13 odd
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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
A level of significance says how much evidence we require to reject Ho in terms of the P-value.The outcome of a test is significant at level if P .
Ex: Determining SignificanceIn ex. Can you balance your checkbook? we examined whether the mean NAEP quantitative scores of young Americans is less than 275. Ho: = 275, Ha: < 275The the z statistic is z = -1.45.
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 = 0.05.
Ex: Is the Screen Tension OK?Recall proper screen tension was 275mV.Is there significant evidence at the 1% level that 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 : = 275 HA : 275(two sided)
No change in the mean tension.There is change in the mean tension.
Step 2: PlanChoose 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 Ha is two sided, we compare = 3.26 with that of /2 = .005 critical value from table C (two tails with total .01).
The critical value is z* = 2.576 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 = 0.01, we reject the null hypothesis at the = 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-valueThe 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) = .001114Knowing 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 IntervalsThe 99% confidence interval for the mean screen tension. 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 soH0 : = 275 is implausible; thus we conclude 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 companiesCourtsMarketersMedical ResearchersReading is fun!
Fixed Significance LevelsChose 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 ) 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.
5% level ( = 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.
Statistical Significance and Practical SignificanceRejection of H0 at the = 0.05 or = 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 TimeTesting 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 = 1.4 days.Reading is fun!
Step 1: StateWe want to test claim about the mean healing time in the population of people treated with NS at the 5% significance level.H0 : = 7.6 mean healing time of scabs is 7.6 daysHa : < 7.6 NS decreases healing time of scabs
Step 2: Plan Since we assume = 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 < = 0.05 we reject Ho 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!