Rejection rules
• It is assumed that Ho is true, aim to test if
this the case or not
• Aim to either not reject or reject Ho with
certain level of confidence (95%, 90%,)
– Can be expressed as (1- a) = 0.95 (or 0.99)
• This helps to define the non-rejection or
rejection region
• identified by critical values (Z or T values)
dependent on level of confidence
– This depends if 1 or 2 tailed test
– See Fig 9.3
1
Critical values v Test statistic
• Need to determine a test statistic to
compare with critical values
– Test stat for mean = x
z
x
• If test statistic falls within non-rejection
region (within boundary of critical values)
then can’t reject Ho
• Accept H1 if test statistic falls in rejection
region
2
Types of errors
• There is a risk of the incorrect conclusion
being made due to sample chosen
• Type 1 error
– =>rejecting Ho when it was in fact true
– Probability of Type 1 error = a
• e.g. for 95% confidence level a = 0.05
• Type 2 error
– => accepting a false hypo, with Prob =
– Size of will depend on hypo value of parameter
3
• Note: a is level of significance (0.05)
• Level of confidence is 1- a = 0.95 or
95%
• Trade off between errors:
– For any sample size anything that
reduces a raises
– A larger sample size could reduce both.
• See Figs 9.12 & 15 and section 9.6
4
Example
• Hubbard’s wants to know with 95% level of
confidence that its cereal boxes contain
more than 500gm
– Past experience suggests that the weight in
cereal boxes is normally distributed.
– Firm takes a random sample of 25 boxes
• Finds
X 520 g
s 75 gm
5
Steps required to conduct Hypo
test
• 1. State hypos
– H0: 500
– H1: > 500 (1 tailed test)
• Note if only concerned if boxes were 500gm then
H1 would be 500
• 2. Select test statistic
– Population is normal, but n
– Use student t dist
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• 3. Calculate critical value based on
level of significance
– = 5% level of significance
– Critical value from t-tables with 24 dof=
– 1.711
• 4. Calculate sample statistic
X
t
s/ n
520 500
1.33
75 / 25
7
• 5. Decision:
– compare test statistic with sample statistic
– as 1.33 falls in non-rejection (acceptance) region
– => do not reject H0
– And conclude that at the 5% level of significance
(or with 95% confidence) the mean fill of cereal
boxes is at least 500gm of cereal.
– If sample statistic = 1.8 then
– Reject H0 and conclude that at this level of
significance there is significant evidence that
boxes contain more than 500gm.
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Alternative approach
• p-value
= observed level of significance
= smallest level at which Ho can be rejected for a given set of data
= area under std normal prob curve outside the test value
given in some computer packages, we wont learn how to
determine it.
• Decision rules:
– If the p-value is a, the null hypo is not rejected
– If the p-value is < a, the null hypo is rejected
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• See Lab 5 for examples
– Need to learn how to conduct Hypo tests for
means, diff in means and proportions
• Next:
– Non-parametric tests
– Ref Black Chap 17 or Keller chap 21.
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