Transcript S2 Hypothesis Testing

The process for setting up a hypothesis test:
State your NULL
e.g. Ho: p = 0.2 or H0 : λ = 4
HYPOTHESIS, H0
State your ALTERNATE
HYPOTHESIS, H1
State the significance
level
State the statistical model
e.g. S.L. = 5% (one or two tailed)
Test your statistic against
the significance level
e.g. P(X > 5)
State your conclusion
in context
e.g. Whether you accept H0 or not
and what this means in context
e.g. H1: p > 0.2 or H1 : λ ≠ 4
e.g. X ~ B(10, 0.2) or X ~ Po(4)
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
State your NULL HYPOTHESIS (H0)
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
H1 : p > 0.25
State your ALTERNATE HYPOTHESIS (H1)
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
H1 : p > 0.25
State your SIGNIFICANCE LEVEL
S.L. : 5% (one tailed)
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
H1 : p > 0.25
X ~ B(20, 0.25)
State the statistical model
S.L. : 5% (one tailed)
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
H1 : p > 0.25
S.L. : 5% (one tailed)
X ~ B(20, 0.25)
P(X ≥ 9) = 1 – P(X ≤ 8)
Test your statistic
= 1 – 0.9591 = 0.0409 = 4.09%
Michelle said she was telepathic. She constructed a simple experiment
to confirm this. She placed four different coloured discs in a bag,
selected one, ensuring her friend could not see it, concentrated on the
colour and then asked her friend to guess the colour. She repeated the
process 20 times and her friend managed to guess correctly 9 times.
Michelle felt this result indicated that she was telepathic. Would you
agree? Test her claim at the 5% significance level.
H0 : p = 0.25
H1 : p > 0.25
S.L. : 5% (one tailed)
X ~ B(20, 0.25)
P(X ≥ 9) = 1 – P(X ≤ 8)
= 1 – 0.9591 = 0.0409 = 4.09%
4.09% is below the 5% significance level
Reject H0 and accept H1, p > 0.25
Evidence to suggest she may be telepathic
Make a conclusion in context
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
State your NULL HYPOTHESIS (H0)
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
H1 : λ < 7.5
State your ALTERNATE HYPOTHESIS (H1)
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
H1 : λ < 7.5
State the significance level
S.L. : 5% (one tailed)
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
H1 : λ < 7.5
X ~ Po(7.5)
State the statistical model
S.L. : 5% (one tailed)
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
H1 : λ < 7.5
X ~ Po(7.5)
P(X ≤ 4) = 0.1321 = 13.21%
Test your statistic
S.L. : 5% (one tailed)
Every year a statistics teacher takes her class out to observe the traffic
passing the school gates during a Tuesday lunch hour. Over the years
she has established that the average number of lorries passing the
gates in a lunch hour is 7.5. During the course of the last 12 months,
a new bypass has been built and the number of lorries passing the
school gates in this year’s experiment was 4. Test, at the 5% level,
whether or not the mean number of lorries passing the school gates
during a Tuesday lunch hour has reduced.
H0 : λ = 7.5
H1 : λ < 7.5
S.L. : 5% (one tailed)
X ~ Po(7.5)
P(X ≤ 4) = 0.1321 = 13.21%
13.21% is above the 5% significance level
Insufficient evidence to reject H0
Mean number of lorries hasn’t reduced
Make a conclusion
In a sack containing a large number of beads ¼ are coloured gold and
the remainder are of different colours. A group of children use some
of the beads in a craft lesson and do not replace them. Afterwards the
teacher wishes to know whether or not the proportion of gold beads
left in the sack has changed. He selects a random sample of 20 beads
and finds that 2 of them are coloured gold.
Stating your hypotheses clearly test, at the 10% level of significance,
whether or not there is evidence that the proportion of gold beads
has changed.
X ~ B(20, 0.25)
S.L. : 5% each side
Two tailed test
H0 : p = 0.25
5%
5%
2
H1 : p ≠ 0.25
P(X ≤ 2) = 0.0913 = 9.13%
5
9.13% is above the 5% significance level
Insufficient evidence to reject H0
Proportion of gold beads hasn’t changed
In a sack containing a large number of beads ¼ are coloured gold and
the remainder are of different colours. A group of children use some
of the beads in a craft lesson and do not replace them. Afterwards the
teacher wishes to know whether or not the proportion of gold beads
left in the sack has changed. He selects a random sample of 20 beads
and finds that 2 of them are coloured gold.
Stating your hypotheses clearly test, at the 10% level of significance,
whether or not there is evidence that the proportion of gold beads
has changed.
X ~ B(20, 0.25)
S.L. : 5% each side
H0 : p = 0.25
5%
5%
5
CRITICAL REGION
The region where H0 would be rejected
H1 : p ≠ 0.25
X ~ B(20, 0.25)
H0 : p = 0.25
S.L. : 5% each side
H1 : p ≠ 0.25
5%
5%
5
P(X ≤ a) = 0.05
a
P(X ≤ 1) = 0.0243
P(X ≤ 2) = 0.0913
b
Critical
X ≤ 1 and X ≥ 9
Region
P(X ≥ b) = 0.05
P(X ≤ b-1) = 0.95
P(X ≤ 7) = 0.8982
P(X ≤ 8) = 0.9591
ACTUAL significance level
0.0243 + (1 – 0.9591)
= 0.0652 or 6.52%
Past records show that 20% of customers who buy crisps from a large
supermarket buy them in single packets. During a particular day a
random sample of 25 customers who had bought crisps was taken
and 2 of them had bought them in single packets.
(a) Use these data to test, at the 5% level of significance, whether or
not the percentage of customers who bought crisps in single packets
that day was lower than usual. State your hypotheses clearly.
At the same supermarket, the manager thinks that the probability of a
customer buying a bumper pack of crisps is 0.03. To test whether or
not this hypothesis is true the manager decides to take a random
sample of 300 customers.
(b) Stating your hypotheses clearly, find the critical region to enable
the manager to test whether or not there is evidence that the
probability is different from 0.03. The significance level should be 5%.
(c) Write down the significance level of this test.
Past records show that 20% of customers who buy crisps from a large
supermarket buy them in single packets. During a particular day a
random sample of 25 customers who had bought crisps was taken
and 2 of them had bought them in single packets.
(a) Use these data to test, at the 5% level of significance, whether or
not the percentage of customers who bought crisps in single packets
that day was lower than usual. State your hypotheses clearly.
S.L. : 5% (one tailed)
X ~ B(25, 0.2)
H0 : p = 0.2
H1 : p < 0.2
P(X ≤ 2) = 0.0982 = 9.82%
9.82% is above the 5% significance level
Insufficient evidence to reject H0
% of people who bought single packets of crisps has not dropped
At the same supermarket, the manager thinks that the probability of a
customer buying a bumper pack of crisps is 0.03. To test whether or
not this hypothesis is true the manager decides to take a random
sample of 300 customers.
(b) Stating your hypotheses clearly, find the critical region to enable
the manager to test whether or not there is evidence that the
probability is different from 0.03. The significance level should be 5%.
np = 9
X ~ B(300, 0.03)
Y ~ Po(9)
H0 : λ = 9
2.5%
2.5%
9
a
P(Y ≤ a) = 0.025
P(Y ≤ 3) = 0.0212
P(Y ≤ 4) = 0.0550
b
S.L. = 2.5% each side
H1 : λ ≠ 9
Critical Region
Y ≤ 3 and Y ≥ 16
P(Y ≥ b) = 0.025
P(Y ≤ b-1) = 0.975
P(Y ≤ 14) = 0.9585
P(Y ≤ 15) = 0.9780
(c) Write down the significance level of this test.
2.5%
2.5%
9
a
P(Y ≤ a) = 0.025
P(Y ≤ 3) = 0.0212
P(Y ≤ 4) = 0.0550
SIGNIFICANCE
LEVEL
Critical Region
X ≤ 3 and X ≥ 16
b
P(Y ≥ b) = 0.025
P(Y ≤ b-1) = 0.975
P(Y ≤ 14) = 0.9585
P(Y ≤ 15) = 0.9780
= 0.0212 + (1 – 0.9780) = 0.0432 = 4.32%
A standard treatment for a drug has a probability of success of 0.3.
A doctor has done some research and has modified the drug and
tested it on 20 patients. He wants to see whether the modified drug
has changed the success rate.
(a) Find the critical region where the significance level needs to be as
close to 5% as possible.
(b) What is the significance level?
(c) 9 patients had success with the modified drug. At the 5% level of
significance, does this give evidence that the drug has improved the
success rate?
(d) On a larger trial using 500 patients, 162 patients had success
with the modified drug. Test at the 2.5% significance level as to
whether there evidence to suggest the drug has improved the success
rate?
A standard treatment for a drug has a probability of success of 0.3.
A doctor has done some research and has modified the drug and
tested it on 20 patients. He wants to see whether the modified drug
has changed the success rate.
(a) Find the critical region where the significance level needs to be as
close to 5% as possible.
X ~ B(20, 0.3)
P(X ≤ a) = 0.025
P(X ≤ 2) = 0.0355
H0 : p = 0.3 H1 : p ≠ 0.3
S.L.: 2.5% each side
P(X ≤ b-1) = 0.975
P(X ≤ 10) = 0.9829
Critical Region: X ≤ 2 and X ≥ 11
(b) What is the significance level?
Significance Level = 0.0355 + (1 – 0.9829) = 0.0526 = 5.26%
(c) 9 patients had success with the modified drug. At the 5% level of
significance, does this give evidence that the drug has improved the
success rate?
Critical Region: X ≤ 2 and X ≥ 11
9 is OUTSIDE the critical region
Insufficient evidence to reject H0
Drug has not improved success rate
(d) On a larger trial using 500 patients, 162 patients had success
with the modified drug. Test at the 2.5% significance level as to
whether there evidence to suggest the drug has improved the success
rate? X ~ B(500, 0.3) np = 150 np(1 – p) = 105 Y ~ N(150, 105)
S.L. = 2.5% one tailed H0 : p = 0.3 H1 : p > 0.3
𝟏𝟔𝟏. 𝟓 − 𝟏𝟓𝟎
= 𝐏 𝒁 > 𝟏. 𝟏𝟐
P(X ≥ 162) ≈ P(Y > 161.5) = 𝐏 𝒁 >
𝟏𝟎𝟓
= 𝟏 − 𝚽(𝟏. 𝟏𝟐) = 𝟏 − 𝟎. 𝟖𝟔𝟖𝟔 = 𝟎. 𝟏𝟑𝟏𝟒 = 𝟏𝟑. 𝟏𝟒%
13.14% is greater than 2.5%, insufficient evidence to
reject H0. The drug has not improved the success rate