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!