Further Statistics theory questions (from 2020-2024 inclusive)
1. State an underlying distribution that is made when using a Wilcoxon signed-rank
test:
a. Data is symmetrically distributed about the median.
2. State why a Wilcoxon rank-sum test might be used:
a. Difference of location test for population not known to be normal
3. Give a reason as to why a Wilcoxon signed-rank test is preferable to a sign test,
when both are valid:
a. Magnitude of differences from median are taken into account.
4. State an assumption necessary for the t-test to be valid:
a. Population is normal.
5. Give two reason why a paired sample t-test may not be valid:
a. Population of differences is not distributed normally/unknown
b. Not a paired sample(depends on the data given in the question)
6. If question is about the mean difference before and after a change in the conditions,
state an assumption that is necessary for this test to be valid:
a. Distribution of population differences is normal.
Hypotheses and conclusions:
1. T-tests
a. Single sample (n<30)
i. Test at the a% significance level whether the sample mean(given by x)
is difference from the population mean:
1. H0: µ=x
2. H1:
a. One tailed: µ<x
b. One tailed: µ>x
c. Two tailed: µ≠x
ii. Comparing statistic with critical value and general conclusion:
1. If test statistic < critical value, accept H0. There is insufficient
evidence to suggest mean is </>/≠ x.
2. If test statistic > critical value, reject H0 and accept H1. There
is sufficient evidence to suggest mean is </>/≠ x.
iii. Exceptions:
1. If there is a belief/claim/suspicion, you must talk about it in the
conclusion, for eg: Insufficient evidence to support the
manager’s claim.
b. Paired sample
i. Test at the a% significance level whether the difference in before and
after means is d
1. H0: Mean difference = d
2. H1:
a. One tailed: Mean difference < d
b. One tailed: Mean difference > d
c. Two tailed: Mean difference ≠ d
ii. Comparing statistic with critical value and general conclusions
1. If test statistic < critical value, accept H0. There is insufficient
evidence to suggest that the mean difference </>/≠ d.
2. If test statistic > critical value, reject H0 and accept H1. There
is sufficient evidence to suggest that the mean difference </>/≠
d.
iii. Exceptions:
1. Always include the claim/belief/suspicion if there is one in the
question in your conclusion.
c. Unpaired sample
i. Test at the a% significance level whether the difference in before and
after means is .
1. H0: µx=µy
2. H1:
a. One tailed: µx>µy
b. One tailed: µx<µy
c. Two tailed: µx≠µy
ii. Comparing statistic with critical value and general conclusion
1. If test statistic < critical value, accept H0. There is insufficient
evidence to suggest that µx >/</≠ µy.
2. If test statistic > critical value, reject H0 and accept H1. There
is sufficient evidence to suggest that µx >/</≠ µy.
iii. What you must not write:
1. Prove/Not sufficient/not enough
iv. Exceptions:
1. Always include the claim/belief/suspicion if there is one in the
question in your conclusion.
2. Chi-squared tests
a. For testing independency
i. Test at the a% significance level whether Event A is independent of
Event B:
1. H0: Event A and Event B are independent.
2. H1: Event A and Event B are not independent
ii. Comparing statistic with critical value and general conclusion:
1. Since test statistic < critical value, accept H0. There is
insufficient evidence to suggest that Event A and Event B are
not independent.
2. Since test statistic > critical value, reject H0 and accept H1.
There is sufficient evidence to suggest that Event A and Event B
are not independent.
iii. What you must not write:
1. Insufficient evidence to show that/prove event A and event B
are independent.
2. Sufficient evidence to show that/prove event A and event B
are independent
iv. Exceptions:
1. If there is a claim/belief/suspicion mentioned in the question,
always conclude with reference to that claim. For example,
if the manager’s claim is that Event A is not independent of
Event B(which is the same as the alternative hypothesis), but
you continue the test and suggest that the null hypothesis is
accepted, you write in the conclusion: Insufficient evidence
to support the manager’s claim
b. Goodness of fit tests
i. Test at the a% significance level where the distribution X is a good fit
for the data:
1. H0: X is a good fit for the data (don’t forget to mention the
parameters of the distribution)
2. H1: X is not a good fit for the data.
ii. Comparing statistic with critical value and general conclusion:
1. Since test statistic < critical value, accept H0. There is
insufficient evidence to suggest that X is not a good fit for the
data (again, don’t forget to include the parameters along with
the distribution)
2. Since test statistic > critical value, reject H0 and accept H1.
There is sufficient evidence to suggest that X is not a good fit
for the data.
iii. What you must not write:
1. Insufficient evidence to show that/prove X is a good fit for
the data.
2. Sufficient evidence to show that/prove X is a good fit for the
data.
iv. Exceptions:
1. If there is a claim/belief/suspicion mentioned in the question,
always conclude with reference to that claim. For example,
if the researcher’s claim is that X is a good fit for the
distribution(which is the same as the null hypothesis), but you
continue the test and suggest that the null hypothesis is
accepted, you write in the conclusion: Sufficient evidence to
support the researcher’s claim.
3. Wilcoxon signed-rank test
a. Single sample
i. Test at the a% significance level whether the data supports the
teacher’s belief (usually saying that the population median = m) or
test at the a% significance level whether there is any evidence against
the teacher’s belief:
1. H0: Population median = m(same as teacher’s belief)
2. H1:
a. One tailed: Population median < m
b. One tailed: Population median > m
c. Two tailed : Population median ≠ m
ii. Comparing statistic with critical value and general conclusion:
1. If test statistic > critical value, accept H0. There is sufficient
evidence to support the teacher’s belief.
2. If test statistic < critical value, reject H0 and accept H1. There
is insufficient evidence to support the teacher’s belief.
iii. What you must not write:
1. Prove/not enough/not sufficient
iv. Exceptions:
1. Sometimes, the belief/claim/suspicion might be contrasting
the null hypothesis.
b. Matched-pairs
i. Test at the a% significance level whether there is any difference in the
medians:
1. H0: Population median for sample 1 = population median for
sample 2
2. H1:
a. One tailed: Population median for sample 1 <
population median for sample 2
b. One tailed: Population median for sample 1 >
population median for sample 2
c. Two tailed: Population median for sample 1 ≠
population median for sample 2
ii. Comparing statistic with critical value and general conclusion
1. If test statistic > critical value, accept H0. There is insufficient
evidence that the median of sample 1 are different from the
median of sample 2.
2. If test statistic < critical value, reject H0 and accept H1. There
is sufficient evidence that the median of sample 1 is </>/≠ the
median of sample 2.
iii. Exceptions:
1. If there is a belief/claim/suspicion, conclusion must be in
context of that. For example, if a person’s claim is the same as
the alternative hypothesis, but the null hypothesis is accepted,
then conclusion must be: insufficient evidence to support the
person’s claim.
4. Wilcoxon rank-sum test
a. Test at the a% significance level whether there is any difference between the
medians of the two samples X and Y:
i. H0: Population median of X = population median of Y
ii. H1:
1. One tailed: Population median of X < population median of Y
2. One tailed: Population median of X > population median of Y
3. Two tailed: Population median of X ≠ population median of Y
b. Comparing statistic with critical value and general conclusion:
i. If test statistic > critical value, accept H0. There is insufficient
evidence to suggest population median of X </>/≠ population median
of Y.
ii. If test statistic < critical value, reject H0 and accept H1. There is
sufficient evidence to suggest population median of X </>/≠
population median of Y.
c. What you must not write:
i. Prove/Not sufficient/Not enough
ii. If conclusion is ‘insufficient evidence to suggest population median of
X </>/≠ population median of Y’ then you must not write ‘sufficient
evidence to suggest population median of X = Y’
d. Exceptions:
i. If there is a belief/claim/suspicion, conclusion must be in context of
that. For example, if a person’s claim is the same as the alternative
hypothesis, but the null hypothesis is accepted, then conclusion
must be: insufficient evidence to support the person’s claim.