,Question 1
Indicate which null hypothesis Sally should specify from the options below:
1. H0: ¯x = 100
2. H 0: µ = 0
3. H0: ¯x > 100
4. H0: µ = 100
Î Answer: The correct answer is option 4.
Sally must compare the mean of her sample of 50 children with a population
mean, and she knows this is equal to 100. A null hypothesis is a hypothesis
which states that there is no effect and in this case it would mean no difference
between the observed mean (calculated after testing the sample of children) and
the population mean of 100. Option 1 is wrong because hypotheses are always
stated in terms of population parameters (not sample statistics), which in the
case of a mean is symbolised by µ. This is also an error in Option 3, but that
error is made worse by the fact that ‘greater than’ is not how we would formulate
a null hypothesis, because ‘greater than’ implies an effect, not a lack of an effect.
Option 2 is wrong because the mean IQ in the population used in the comparison
is 100, not 0.
Question 2
From reading the psychometric test manual, Sally knows that the IQ test was
standardized on a mean of 100 and a standard deviation of 15. Which of the
options below would be the most appropriate statistical test which she could use
to test the hypotheses?
1. tc
2. zx¯
3. tx¯
4. td
Î Answer: Option 2 gives the most appropriate test statistic to use.
Sally needs to compare a sample mean with a constant population mean which
is already know to be 100.
This implies a single sample test for comparing a mean with a given value (100
in this case). It would therefore be either a single sample z-test or a single sample
t-test. Because the population standard deviation is known (it is specified in the
question above as σ = 15) the most appropriate test here would be the single
sample z-test, which is written z¯x. Option 1 and option 4 both refer to two-
sample t-tests which are to be used to compare two sample means and not one
sample mean with a constant population mean.
Question 3
Sally decides to perform her statistical test at a level of significance of α = 0.05.
Based on the data from the gifted children the computer calculates a test statistic
and reports a two-tailed p-value of p = 0.082. What can Sally conclude with
regard to the hypotheses?
,1. She should reject H0 which implies that the IQs of the gifted children is
significantly higher than the population average
2. She cannot reject H0 which implies that she cannot conclude that the IQs
of the gifted children is significantly higher than the population average
3. She should reject H1 which implies that the IQs of the gifted children is
not higher than the population average
4. She cannot make a conclusion because the exact value of the calculated
test statistic is not provided
In this question, the words ‘which implies that’ should be interpreted as
referring to the consequence of rejecting the null hypothesis or not rejecting it.
To make this more clear, we put an alternative formulation of the question on
myUnisa. Note that this is the same question so the answer is not affected.
The alternative formulation is as follows:
Sally decides to perform her statistical test at a level of significance of α = 0.05.
Based on the data from the gifted children the computer calculates a test statistic
and reports a two-tailed p-value of p = 0.082. What can Sally conclude with
regard to the hypotheses?
1. She should reject H0, and can conclude that the IQs of the gifted children
is significantly higher than the population average
2. She cannot reject H0 and so she cannot conclude that the IQs of the gifted
children is significantly higher than the population average
3. She should reject H1 and can therefore conclude that the IQs of the gifted
children is not higher than the population average
4. She cannot make a conclusion because the exact value of the calculated
test statistic is not provided
Î Answer: The correct answer is option 1.
To test whether the null hypothesis should be rejected, the calculated p-value
should be found to be smaller than the level of significance α, which was set by
the researcher at 0.05. This would imply that the probability of making an error
if we reject the null hypothesis is less than α, which is why we would be willing
to reject it. Note however that the alternative hypothesis being tested is one
sided. This is implied in the scenario above: Sally wants to know whether the
mean IQ score of children in the school for gifted children is higher than the
population average of 100. She is not interested in the outcome if the mean IQ
score is significantly less than this population mean. The two-tailed p-value
should therefore be divided by 2 to calculate a one-sided p-value:
One-tailed p-value = (two-tailed p-value)/2 = 0.082/2 = 0.041 (see p.
81 of the PYC3704 Guide).
This one-tailed p-value is indeed smaller than the chosen level of significance
(p-value = 0.041 < α = 0.05), so the null hypothesis can be rejected. This implies
that the mean IQ of the gifted children is significantly higher than the population
average of 100.
Note that the reason why we calculate a test statistic is to use it to determine
what the p-value would be. In this case the p-value was given (even though we
needed to adjust it), so Option 4 is not valid.
, Question 4
When applying a statistical test, the probability of a Type I error is equal to - - -
- -.
1. the level of significance
2. the p-value of the test statistic under the null hypothesis
3. the p-value of the test statistic under the alternative hypothesis
4. 0.05 or 0.01
Î Answer: Option 2 is correct.
An error of Type I is when a researcher makes an error by rejecting the null
hypothesis when it should not be rejected, and the p-value gives the probability
that this error will be made based on the specific data being observed (see
section 3.3.2 of the PYC3704 Guide). That is to say, the p-value is calculated
with the assumption that the null hypothesis is true (that is what ‘under the null
hypothesis’ means) and this possibility is compared with the sample data (in this
case, the calculated sample mean). So what is being investigated is the
probability that any effect being observed in the data is merely the consequence
of chance effects such as measurement error, and this directly reflects the size
of the error which will be made if the null hypothesis is rejected (an error of
Type I). A researcher would want to keep this value small (smaller than α).
Option 1 is wrong because the level of significance is the maximum probability
of making an error of type I that we would allow – that is, we want the p-value
to fall below this – but it is not a direct estimate of that error for a specific
observation of a sample of data. We actually should choose this value before
making observations by looking at a sample of data. Option 3 is incorrect
because it is the probability of the null hypothesis being true that is investigated,
not the alternative hypothesis. Type I errors are based on calculated p-values for
a specific set of data, not predetermined to be 0.05 or 0.01 (unlike the level of
significance α), so Option 4 is also wrong.
Question 5
The lower we set the level of significance, the greater the probability of - - - - -
1. rejecting the null hypothesis
2. a type I error
3. a type II error
4. accepting the alternative hypothesis
Î Answer: Option 3 is correct.
We know that the extent of the type I error that a researcher is willing to make
is controlled by the researcher by setting the level of significance (α) in advance.
The probability of a type II error (β) is not controlled in advance by the
researcher except for the fact that we know that the lower (smaller) the
probability of a type I error (α) the greater (larger) the probability of a type II
error (β). You could eliminate error of type I (rejecting H0 when you should not)
altogether by never rejecting the null hypothesis, irrespective of how small the
p-value becomes, but that would make the probability of not rejecting H0 when
you should reject it (error of type II) an absolute certainty. See page 83 in the
Guide for PYC3704.
Indicate which null hypothesis Sally should specify from the options below:
1. H0: ¯x = 100
2. H 0: µ = 0
3. H0: ¯x > 100
4. H0: µ = 100
Î Answer: The correct answer is option 4.
Sally must compare the mean of her sample of 50 children with a population
mean, and she knows this is equal to 100. A null hypothesis is a hypothesis
which states that there is no effect and in this case it would mean no difference
between the observed mean (calculated after testing the sample of children) and
the population mean of 100. Option 1 is wrong because hypotheses are always
stated in terms of population parameters (not sample statistics), which in the
case of a mean is symbolised by µ. This is also an error in Option 3, but that
error is made worse by the fact that ‘greater than’ is not how we would formulate
a null hypothesis, because ‘greater than’ implies an effect, not a lack of an effect.
Option 2 is wrong because the mean IQ in the population used in the comparison
is 100, not 0.
Question 2
From reading the psychometric test manual, Sally knows that the IQ test was
standardized on a mean of 100 and a standard deviation of 15. Which of the
options below would be the most appropriate statistical test which she could use
to test the hypotheses?
1. tc
2. zx¯
3. tx¯
4. td
Î Answer: Option 2 gives the most appropriate test statistic to use.
Sally needs to compare a sample mean with a constant population mean which
is already know to be 100.
This implies a single sample test for comparing a mean with a given value (100
in this case). It would therefore be either a single sample z-test or a single sample
t-test. Because the population standard deviation is known (it is specified in the
question above as σ = 15) the most appropriate test here would be the single
sample z-test, which is written z¯x. Option 1 and option 4 both refer to two-
sample t-tests which are to be used to compare two sample means and not one
sample mean with a constant population mean.
Question 3
Sally decides to perform her statistical test at a level of significance of α = 0.05.
Based on the data from the gifted children the computer calculates a test statistic
and reports a two-tailed p-value of p = 0.082. What can Sally conclude with
regard to the hypotheses?
,1. She should reject H0 which implies that the IQs of the gifted children is
significantly higher than the population average
2. She cannot reject H0 which implies that she cannot conclude that the IQs
of the gifted children is significantly higher than the population average
3. She should reject H1 which implies that the IQs of the gifted children is
not higher than the population average
4. She cannot make a conclusion because the exact value of the calculated
test statistic is not provided
In this question, the words ‘which implies that’ should be interpreted as
referring to the consequence of rejecting the null hypothesis or not rejecting it.
To make this more clear, we put an alternative formulation of the question on
myUnisa. Note that this is the same question so the answer is not affected.
The alternative formulation is as follows:
Sally decides to perform her statistical test at a level of significance of α = 0.05.
Based on the data from the gifted children the computer calculates a test statistic
and reports a two-tailed p-value of p = 0.082. What can Sally conclude with
regard to the hypotheses?
1. She should reject H0, and can conclude that the IQs of the gifted children
is significantly higher than the population average
2. She cannot reject H0 and so she cannot conclude that the IQs of the gifted
children is significantly higher than the population average
3. She should reject H1 and can therefore conclude that the IQs of the gifted
children is not higher than the population average
4. She cannot make a conclusion because the exact value of the calculated
test statistic is not provided
Î Answer: The correct answer is option 1.
To test whether the null hypothesis should be rejected, the calculated p-value
should be found to be smaller than the level of significance α, which was set by
the researcher at 0.05. This would imply that the probability of making an error
if we reject the null hypothesis is less than α, which is why we would be willing
to reject it. Note however that the alternative hypothesis being tested is one
sided. This is implied in the scenario above: Sally wants to know whether the
mean IQ score of children in the school for gifted children is higher than the
population average of 100. She is not interested in the outcome if the mean IQ
score is significantly less than this population mean. The two-tailed p-value
should therefore be divided by 2 to calculate a one-sided p-value:
One-tailed p-value = (two-tailed p-value)/2 = 0.082/2 = 0.041 (see p.
81 of the PYC3704 Guide).
This one-tailed p-value is indeed smaller than the chosen level of significance
(p-value = 0.041 < α = 0.05), so the null hypothesis can be rejected. This implies
that the mean IQ of the gifted children is significantly higher than the population
average of 100.
Note that the reason why we calculate a test statistic is to use it to determine
what the p-value would be. In this case the p-value was given (even though we
needed to adjust it), so Option 4 is not valid.
, Question 4
When applying a statistical test, the probability of a Type I error is equal to - - -
- -.
1. the level of significance
2. the p-value of the test statistic under the null hypothesis
3. the p-value of the test statistic under the alternative hypothesis
4. 0.05 or 0.01
Î Answer: Option 2 is correct.
An error of Type I is when a researcher makes an error by rejecting the null
hypothesis when it should not be rejected, and the p-value gives the probability
that this error will be made based on the specific data being observed (see
section 3.3.2 of the PYC3704 Guide). That is to say, the p-value is calculated
with the assumption that the null hypothesis is true (that is what ‘under the null
hypothesis’ means) and this possibility is compared with the sample data (in this
case, the calculated sample mean). So what is being investigated is the
probability that any effect being observed in the data is merely the consequence
of chance effects such as measurement error, and this directly reflects the size
of the error which will be made if the null hypothesis is rejected (an error of
Type I). A researcher would want to keep this value small (smaller than α).
Option 1 is wrong because the level of significance is the maximum probability
of making an error of type I that we would allow – that is, we want the p-value
to fall below this – but it is not a direct estimate of that error for a specific
observation of a sample of data. We actually should choose this value before
making observations by looking at a sample of data. Option 3 is incorrect
because it is the probability of the null hypothesis being true that is investigated,
not the alternative hypothesis. Type I errors are based on calculated p-values for
a specific set of data, not predetermined to be 0.05 or 0.01 (unlike the level of
significance α), so Option 4 is also wrong.
Question 5
The lower we set the level of significance, the greater the probability of - - - - -
1. rejecting the null hypothesis
2. a type I error
3. a type II error
4. accepting the alternative hypothesis
Î Answer: Option 3 is correct.
We know that the extent of the type I error that a researcher is willing to make
is controlled by the researcher by setting the level of significance (α) in advance.
The probability of a type II error (β) is not controlled in advance by the
researcher except for the fact that we know that the lower (smaller) the
probability of a type I error (α) the greater (larger) the probability of a type II
error (β). You could eliminate error of type I (rejecting H0 when you should not)
altogether by never rejecting the null hypothesis, irrespective of how small the
p-value becomes, but that would make the probability of not rejecting H0 when
you should reject it (error of type II) an absolute certainty. See page 83 in the
Guide for PYC3704.
