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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. - PowerPoint PPT Presentation
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Tests with Fixed Tests with Fixed Significance LevelSignificance LevelTarget Goal:Target Goal:I can reject or fail to reject the null I can reject or fail to reject the null hypothesis at different significant hypothesis at different significant levels.levels.I can determine how practical my I can determine how practical my results are.results are.
9.1b
h.w: pg 546: 9 – 13 odd
• 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.
Ex: Ex: Determining SignificanceDetermining 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 = -1.45.
Is this evidence against HIs this evidence against Hoo statistically statistically
significant at the 5% level?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.
Ex: Ex: Is the Screen Tension OK?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(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:Step 3: Do -Do - If the conditions are met, If the conditions are met, carry out the inference procedure.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).
,/
xz
n
306.3 275
3.2643/ 20
z
z
• 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
This does not tell us a lot. 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).
Test from Confidence IntervalsTest from Confidence Intervals
The 99% confidence interval for the mean screen tension.
• μ is =
= (281.5, 331.1)
Or, STAT:Tests:ZInterval:Stats (try!)
*x zn
43306.3 2.576
20
• 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.
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 LevelsFixed 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.
• 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.
Statistical Significance and Practical Statistical Significance and Practical SignificanceSignificance
• 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!
Ex. 1 WEx. 1 Wound Healing Timeound 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: StateStep 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
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:Step 3: Do Do Compute the test statistic and find the p – value.Compute the test statistic and find the p – value.
Standardize: P( < 7.6)
= P(Z < -1.79) = .0367
x
7.1 7.6 <
1.4 25( 7.6)xP P z
Step 4:Step 4: Interpret your results in the Interpret your results in the context of the problem.context of the problem.
• Since our p value, .0367 < α = 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!