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Summary Statistics 1B
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summary of the book introduction to the practice of statistics
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Statistic 6.2Tests of significance:
The confidence interval is appropriate when our goal is to
estimate population parameters
Tests of significance – to assess the evidence provided by the
data in favor of some claim about the population parameter
o a formal procedure for comparing observed data with a
hypothesis whose truth we want to asses
o a process of assessing the significance of the evidence
provided by data against the NH
o Step 1: state the NH and the AH
o Step 2: calculate the value of the test statistic
o Step 3: find the p-value
o Step 4: state a conclusion; e.g. a significance level ∝
hypothesis is a statement about the population parameters
the results of a test are expressed in terms of a probability
that measures how well the data and the hypothesis agree
Stating Hypotheses:
null hypothesis – the statement being tested in a test of
significance; usually the null hypothesis is a statement of “no
effect” or “no difference in the true means”
o e . g . H 0 : there is no difference in the true means
o a statement about the population parameter
o refers to the true means
alternative hypothesis – name of the statement that we hope
or suspect to be true instead of the null hypothesis
o e . g . H a : the true means are not the same
o is the statement that we will accept if the evidence
enables us to reject the NH
the test of significance is designed to asses the strength of the
evidence against the null hypothesis
“if” the previous sentence is true; the null hypothesis is true
then ask whether the data provide evidence against the
supposition we have made
Step 1: state a claim that we will try to find evidence against
Hypothesis always refer to some populations not to a
particular outcome; therefore we always have to state our NH
and AH in terms of population parameters
Alternative hypothesis can whether be one-sided or two-sided;
which refers whether a parameter differs from its null
hypothesis value in a specific direction or in either direction
Test statistics:
, When NH is true, we expect the estimate to take a value near
the parameter value specified by NH
Values of the estimate far away from the parameter value
specified by NH give evidence against NH; the alternative
hypothesis determines which directions count against NH
To assess how far the estimate is from the parameter,
standardize the estimate
estimate−hypothesized value
o z=
standard deviationof the estimate
a test statistic measures compatibility b/w the NH and the
data
o usually measures how far the data are from the NH
557−0
e.g. z= =1.49→ we have observed a sample estimate
374
that is about one and a half standard deviations away from
the hypothesized value of the parameter
p-values:
two or three standard deviations is its criterion for rejecting
NH
a test of significance finds the probability of getting an
outcome as extreme or more extreme than the actually
observed outcome
extreme = far from what we would expect
the direction or directions that count as far from what we
would expect are determined by the NH and AH
p-value – the probability, assuming the NH is true, that the
test statistic would take a values as extreme or more extreme
than that actually observed
o the smaller the p-value, the stronger the evidence
against the NH provided by the data
o e.g. p.362 6.12
Statistical significance:
we can compare the p-value we calculated with a fixed value
the we regard as decisive entscheidend; this amounts announce
in advance how much evidence against the NH is required to
reject the NH
significance level – the decisive value of p, commonly know as
α
∝=0.05
if the p-value is less or equal to ∝ , you conclude that the AH
is true; if the p-value is greater than ∝ , than you conclude
that the data do not provide sufficient evidence to reject the
NH
Tests for a population mean:
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