Chapter 11: Testing a Claim
Confidence intervals are one of the two most common types of
statistical inference. We use confidence intervals when we wish
to estimate a population parameter. The second type of
inference is called significance tests. The goal here is to assess
the evidence provided by data about some claim concerning the
population.
Example: I claim I can make 80% of my basketball free throws.
We could test this by collecting data on free throws and seeing if the %
is close to the claim. If it is not, we can determine if the % if “far
enough” away from the claim to reject that original claim.
Summary of Significance Testing
1) Hypothesis about a population parameter (m or p)
2) Check conditions for testing
3) Collect data to find a test statistic
4) Find a probability that extreme values are possible (p-value)
5) Look at the significance level and interpret the results of the
claim
6) The reasoning is that we are looking at what will happen in
the long run with repeated sampling or experiments.
Significant = not likely to happen by chance
Stating the Hypothesis
Null Hypothesis (H0): the statement being tested in a
significance test (“no effect”, “no difference”, “no change”)
We are trying to find evidence AGAINST this claim
Alternative Hypothesis (Ha): the claim about the population
that we are trying to find evidence FOR.
These are both stated BEFORE the data (evidence) is provided.
Ha > H0 value
Ha < H0 value
Ha can also be ≠
(this makes the significance testing two-sided)
Conditions for Significance Testing
1) SRS
2) Normality
• For means: population distribution is Normal or n ≥ 30
(CLT)
• For proportions: npˆ  10
and
n(1  pˆ )  10
3) Independence
As with any type of inference, you MUST check the
conditions and state the reasoning for each condition
EVERY TIME!!
Test Statistic
•Collect data and calculate x or p̂ and standard error
•We want to compare the sample’s calculations with H0, so we
compute a test statistic (z-score)
test statistic (z) = estimate ( x or p̂ ) – H0 value
standard error
But how large is large enough to prove H0 untrue?
P-value: the probability of getting another sample statistic as or
more extreme than the current estimate when H0 is assumed true
…the probability that the current estimate is a “fluke”
The smaller the P-value, the stronger evidence AGAINST H0
To determine if the P-value is “good enough”, we compare that
probability to the a significance level (a).
a= the maximum P-value to still provide evidence AGAINST H0
Significance levels, like confidence levels, are chosen at the
discretion of the statistician depending on the situation.
a typically is set at 0.01, 0.05, or 0.10
We want P ≤ a
1) Hypothesis: Let’s assume the mean times did NOT decrease
H0: m = 6.7 minutes
Ha: m < 6.7 minutes
2) Conditions:
- randomness: SRS of 400 was collected
- Normality: by CLT this distribution will also be normal and
standard error will be

2

 0.10
n
400
3) Test Statistic:
6.48  6.7
z
 2.20
2 / 400
If z is large and in the
direction of Ha, then it is
unlikely that H0 is true.
To find the P-value, look up the test statistic (z) in the table
z=-2.20  0.0139
Compare to a significance level…
For a = 0.01, our P-value of 0.0139 would NOT be significant
(ie: the 6.48 could be a fluke)
For a = 0.05, our P-value of 0.0139 WOULD be significant
(ie: the 6.48 is mostly likely NOT a fluke and there
is a significant decrease in response time)
The significance level should be determined BEFORE you collect
the data and calculate a test statistic. (At the same time you
create the hypotheses.)
Interpreting the Results/Draw a Conclusion
P≤a
P>a
we reject H0
we fail to reject H0
If H0 is true, the
sample results are too
unlikely to occur by
chance
The sample results
could possibly occur by
chance
This does not
mean we accept
Ha
This does not
mean we accept
H0