Download 9.1b Tests w fixed sig. level

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 Ho 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 = -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
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.
• H0 : μ = 275 No change in the mean tension.
• HA : μ ≠ 275 There is change in the mean tension.
(two sided)
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
x 
306.3  275
z
, z
 3.26
/ n
43/ 20
• Determine significance at the 1% level
Because Ha is two sided, we compare z =
3.26 with that of α/2 = .005 critical value
from table C (two tails with total .01).
• The critical value is z* = 2.576
invNorm(.995)
3.26
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.
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 Intervals
The 99% confidence interval for the mean
screen tension.
43

• μ is = x  z *
 306.3  2.576
20
n
= (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
• 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 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
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!
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.
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.
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
Step 2: Plan
Since we assume σ = 1.4 days, use a onesample 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( x < 7.6)

7.1  7.6 
P( x 7.6)  P  z <

1.4 25 

= 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 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!