Transcript 9.1a Tests of Significance

Chapter 9
Tests of Significance
Target Goal:
I can perform a significance test to support the
alternative hypothesis.
I can interpret P values in context.
9.1a Testing a Claim
h.w: pg. 546: 1-7 odd
Tests of Significance: to test a claim about an
unknown population parameter.
Example : Sweetening Colas
• Diet colas use artificial sweeteners to avoid sugar.
Colas with artificial sweeteners gradually lose
sweetness over time. Manufacturers therefore test
new colas for loss of sweetness before marketing
them. Trained tasters sip the cola along with
drinks of standard sweetness and score the cola on
a “sweetness score” of 1 to 10. The cola is then
stored for a period of time, then each taster scores
the stored cola.
• This is a matched pairs experiment.
• The reported data is the difference in tasters’
scores. The bigger the difference, the bigger the
loss in sweetness.
2.0 0.4 0.7 2.0 -0.4
2.2 -1.3 1.2 1.1 2.3
Positive score: lost sweetness
Negative score: gained sweetness
• The sample mean indicates a small loss of
sweetness. x  1.02
• Consider that a different sample of tasters
would have resulted in different scores, and
that some variation in scores is expected
due to chance.
Does the data provide good evidence that
the cola lost sweetness in storage?
Perform a
Significance Test.
Step 1: State - Identify the population
parameter. State the null and alternative
hypothesis.
• The parameter of interest is μ, the mean
loss in sweetness of all colas produced by
the manufacturer.
.
There is no effect or change in the population.
This is the statement we are trying to find evidence
against.
• The cola does not lose sweetness.
H 0 :   0 “ H- nought or null hypothesis”
There is an effect or change in the population.
• This is the statement we are trying to find
evidence for.
• The cola does lose sweetness.
H a :   0 alternative hypothesis
Step 2: Plan
Choose the appropriate inference procedure.
Verify the conditions for using the selected
procedure.
• We will be more specific about this later.
Step 3: Do - Calculate a statistic
to estimate the parameter.
Is the value of the statistic far from the
value of the parameter?
• If so, reject the null hypothesis.
• If not, accept the null hypothesis.
Calculate the test statistic.
• Suppose the individual tasters’ scores vary
according to a normal distribution with
mean μ and σ = 1.
• We want to test the null hypothesis so we
assume μ = 0.
• The sampling model for x is approximately
normal with mean   0 and standard
deviation

1
n

10
 0.316
N(0, 0.316)
Compare one cola with x = 0.3 and our cola
with x = 1.02 to show what it means in
terms of H 0 :   0 .
-0.9 -0.6 -0.3 0 0.3 0.6 0.9
The cola with 0.3 could happen just by chance.
Our cola is so far out on the normal curve that it is
good evidence that this cola did lose sweetness.
Reject the null hypothesis.
Our sample mean, x , was 1.02.
• Assuming that the null hypothesis is true, what
is the probability of getting a result at least that
large?
Calculate the P-value.
(the probability of the observed x )
• If P-value is small, your result is statistically
significant.
Determine P-value
• Normalcdf ( 1.02, 1E99, 0, 0.316 ) = 0.0006
• The probability to the right of x is called the
P-value.
• The P-value is small, so our result is statistically
significant. Reject the null hypothesis.
0.0006
1.02
Alternate way to calculate one sided P-Value
Find Z-value for N(0, 0.316 ), for sample mean 0.3.
 x  0 0.3  0 

P( x  0.3)  P 

 0.316 0.316 
• = P(Z ≥ 0.95)
•
= 1 - P(Z ≤ 0.95)
•
= 1 – 0.8289 = 0.1711
Is this good evidence against the null hypothesis?
• There is about a 17% chance that we will
obtain a sample of 10 sweetness loss values
whose mean is 0.3 or greater.
• This sample could occur quite easily by
chance alone.
• This evidence against Ho is not strong.
(High P-value)
Step 4: Interpret the results in the context of
the problem for our original data.
x  1.02
• The P-value is small,0.0006 (good evidence
against null hypothesis).
• meaning that we would only expect to get
this result in 6 out of 10,000 samples.
• This is very unlikely, so we will reject the
null hypothesis in favor of the alternative
hypothesis and conclude that the cola
actually did lose sweetness.
Statistically Significant
• If the P-value is small we say that our result is
statistically significant.
• The smaller the P-value, the stronger the
evidence provided by the data.
How small is small enough?
• Compare the P-value to the value of the
significance level α (alpha). This value is usually
predetermined.
• If the P-value is as small as or smaller than α, we
say that the data are statistically significant at
level α.
Read pg. 527 - 532