The set of values of the test
The probability of getting an outcome
statistic which lead to rejection at least as extreme as that observed,
if the null hypothesis is true.
of the null hypothesis.
Null hypothesis, symbol H 0
Test statistic
Alternative hypothesis,
symbol H 1
A value which gives a characteristic
of the population. Often symbolised
by a Greek letter.
A test which looks for evidence
that the parameter is greater than
(or less than) a particular value.
Critical value
Critical region
A single value calculated from the sample.
It is used to make a decision.
The belief you start with.
You will only stop believing
this if there is enough evidence.
Significance level
The probability that the null hypothesis
is rejected, even though it is true.
Parameter
The value which you compare
the test statistic with to decide
whether to reject the null hypothesis.
This is what you are
looking for evidence of.
p value
1
tail test
The belief you start with.
Null hypothesis, symbol H 0 You will only stop believing
this if there is enough evidence.
Alternative hypothesis,
symbol H 1
This is what you are
looking for evidence of.
p value
The probability of getting an outcome
at least as extreme as that observed,
if the null hypothesis is true.
Significance level
The probability that the null hypothesis
is rejected, even though it is true.
Test statistic
A single value calculated from the sample.
It is used to make a decision.
Parameter
A value which gives a characteristic
of the population. Often symbolised
by a Greek letter.
Critical value
The value which you compare
the test statistic with to decide
whether to reject the null hypothesis.
Critical region
The set of values of the test
statistic which lead to rejection
of the null hypothesis.
1
tail test
A test which looks for evidence
that the parameter is greater than
(or less than) a particular value.
3 Over a long period of time, 20% of all bowls made by a particular manufacturer
are imperfect and cannot be sold.
The manufacturer introduces a new process for producing bowls. To test whether
there has been an improvement, each of a random sample of 20 bowls made by
the new process is examined.
From this sample, 2 bowls are found to be imperfect.
(ii) Show that this does not provide evidence, at the 5% level of significance, of a
reduction in the proportion of imperfect bowls. You should show your hypotheses
and calculations clearly.
[6]
(MEI S1 Jan 2006 (part))
State the null hypothesis
State the alternative hypothesis
Say what p stands for
H 0 : p = 0.2
H1: p < 0.2
p is the proportion of imperfect bowls
produced
Decide what the distribution seen in the X ∼ B(20, 0.2) where X is the number of
sample would be if the null hypothesis
imperfect bowls in the sample
is true
Decide whether large or small values of Small values of X would lead to
X (or both) would lead to rejection of
rejection of H 0
the null hypothesis
P( X ≤ 2) = 0.2061
Find the probability of the observed
value of X and the values more
extreme
Compare the probability to the
20.61% > 5%
significance level
Decide whether to accept or reject the
Accept H 0
null hypothesis
State the decision in a way that relates There is insufficient evidence, at the
to the original situation
5% level of significance, of a reduction
in the proportion of imperfect bowls
Mark scheme
3
(i)
(ii)
X ~ B(10,0.2)
P(X < 4) = P(X ≤ 3) = 0.8791
OR attempt to sum P(X = 0,1,2,3) using
X ~ B(10,0.2) can score M1, A1
Let p = the probability that a bowl is imperfect
H 0 : p = 0.2
H1: p < 0.2
X ~ B(20,0.2)
P(X ≤ 3) = 0.2061 0.2061 > 5%
M1 for X ≤ 3
A1
2
B1 Definition of p
B1, B1
Cannot reject H 0 and so insufficient evidence
B1 for 0.2061 seen
M1 for this comparison
to claim a reduction.
A1 dep for comment in context
OR using critical region method:
CR is {0} B1, 2 not in CR M1, A1 as above
TOTAL
Examiners’ report
In the hypothesis test, although many candidates gave correct hypotheses in
terms of p, few defined p explicitly in words. Centres should advise candidates
that such a definition does attract credit. It was notable that from any given
centre it was usually the case that either almost all candidates defined p or no
candidates did so. The hypotheses themselves were usually correctly given but a
number of candidates still continue to lose marks through poor notation.
Candidates should be aware that H0 = 0.2 is not an acceptable notation, nor is H0
: P(X=0.2). The standard notation is H0: p = 0.2. As in previous sessions, many
candidates used point probabilities, which effectively prevents any further credit
being gained. Those who were successful in comparing the tail probability of
0.2061 with 0.05 often lost the final mark by not putting their conclusion in
context. To simply state ‘Accept H0’ on its own is not sufficient to gain credit
here. A conclusion along the lines of ‘There is insufficient evidence to claim that
there has been a reduction’ is needed to gain the mark.
An argument based on critical regions is of course perfectly acceptable, but
candidates preferring to use such arguments need to be very precise. To simply
state that the critical region is {0} without a probability justification is
insufficient.
3
3
8
Accept H 0
p is the proportion of
imperfect bowls produced
State the null hypothesis
Decide whether large or small values
of X (or both) would lead to
rejection of the null hypothesis
State the alternative hypothesis
P(X ≤ 2) = 0.2061
There is insufficient evidence, at the
5% level of significance, of a reduction
in the proportion of imperfect bowls
Compare the probability
to the significance level
Decide whether to accept Small values of X would
or reject the null hypothesis lead to rejection of H 0
H 0 : p = 0.2
Decide what the distribution
seen in the sample would be
if the null hypothesis is true
X ∼ B(20, 0.2)
where X is the number of
imperfect bowls in the sample
Find the probability of the
observed value of X and
the values more extreme
20.61% > 5%
H 1 : p < 0.2
Say what p stands for
State the decision in a way that
relates to the original situation
Dream Number
In the National Lottery Dream Number game seven digits are drawn at random. In
the 17 draws in November and December 2007, the first digit to be drawn was 2 on
four occasions. If the draw is truly random, the probability of the first digit being 2
should be
random?
1
. Is there evidence, at the 5% level of significance, that the draw is not
10
Three solutions are given below; each of them is incorrect or incomplete (or both).
What mistakes have been made in each solution?
Can you produce a correct and complete solution?
4
= 23.5% .
17
The probability should be
1
= 10% .
10
There is more than 5% difference between these so there is evidence that the draw is not random.
H 0 : p = 0.1
H1 : p ≠ 0.1
X ~ B(7, 0.1)
P( X ≥ 4)
P( X ≥ 4) = 1 − P( X ≤ 3) = 1 − 0.9973 = 0.0027 = 0.27%
!
X ~ B(17, 0.1)
P( X ≥ 4)
P( X ≥ 4) = 1 − P( X ≤ 4) = 1 − 0.9779 = 0.0221 = 2.21%
!
!