Level I R11 Hypothesis Testing
Test Code: L1 R11 HYPT Q-Bank 2020
Number of questions: 36
Question
1
Q-Code: L1-QM-HYPT-001
LOS a
Section 2
LOS a
Section 2
Which of the following is most likely to be the first step in hypothesis testing?
A) Specifying the significance level.
B) Stating the decision rule.
C) Stating the hypothesis.
Answer
C) Stating the hypothesis.
Explanation C is correct. The first step in hypothesis testing is ‘stating the hypothesis’.
In order to test a hypothesis, we follow these steps:
1. Stating the hypothesis.
2. Identifying the appropriate test statistic and its probability distribution.
3. Specifying the significance level.
4. Stating the decision rule.
5. Collecting the data and calculating the test statistic.
6. Making the statistical decision.
7. Making the economic or investment decision.
Section 2. LO.a.
Question
2
Q-Code: L1-QM-HYPT-003
Professor Alan Chang is reviewing the following statements made by his students:
Beth: The null hypothesis is the hypothesis that is being tested; and a two tailed hypothesis may have either of
the two signs: $leq$ or $geq$.
Donald: Specifying the significance level, $alpha$, isn't a necessary step and one could do without it during
hypothesis testing.
Kevin: The test statistic is a quantity calculated based on a sample, whose value is the basis for deciding whether
or not to reject the alternate hypothesis.
Whose statements will Professor Chang will least likely agree to?
A) Only Donald.
B) Only Donald and Beth.
C) All of them.
Answer
C) All of them.
Explanation C is correct.
Beth is incorrect because the null hypothesis is the hypothesis that is tested, and a two tailed hypothesis has the
sign: =.
Donald is incorrect because specifying the significance level, α, is a necessary step and one cannot do without it
during hypothesis testing.
Kevin is incorrect because the test statistic is a quantity calculated based on a sample, whose value is the basis for
deciding whether or not to reject the null hypothesis. Section 2. LO.a.
Question
3
Q-Code: L1-QM-HYPT-005
Which of the following statement (s) is most likely correct regarding a Type II error?
A) Type II error fails to reject a false null hypothesis.
B) Type II error fails to reject a true null hypothesis.
C) Type II error rejects a true null hypothesis.
LOS c
Section 2
Answer
A) Type II error fails to reject a false null hypothesis.
Explanation A is correct. In reaching a statistical decision, we can make two possible errors:
Type I error: We may reject a true null hypothesis.
Type II error: We may fail to reject a false null hypothesis.
The following table shows the possible outcomes of a test.
Decision
True condition
H0 true
Do not reject H0
Reject H0 (accept Ha )
H0 false
Correct decision
Type II error
Type I error
Correct decision
Section 2. LO.c.
Question
4
Q-Code: L1-QM-HYPT-007
LOS c
Section 2
LOS d
Section 2
When a false null hypothesis is not rejected, it leads to a/an:
A) Type I Error.
B) Type II Error.
C) acceptance of the alternate hypothesis.
Answer
B) Type II Error.
Explanation B is correct. In reaching a statistical decision, we can make two possible errors:
Type I error: We may reject a true null hypothesis.
Type II error: We may fail to reject a false null hypothesis.
The following table shows the possible outcomes of a test.
Decision
True condition
H0 true
H0 false
Do not reject H0
Correct decision
Type II error
Reject H0 (accept Ha )
Type I error
Correct decision
Section 2. LO.c.
Question
5
Q-Code: L1-QM-HYPT-009
Jane Norah is an analyst for a midcap growth fund. The fund earns a quarterly return of 4.5 percent relative to an
estimated return of 6.0 percent. If Norah wishes to test whether the actual results are different from the estimated
return of 6 percent, the null hypothesis is most likely:
A) H0: µ ≤ 6.0.
B) H0: µ = 6.0.
C) H0: µ ≠ 6.0.
Answer
B) H0: µ = 6.0.
Explanation B is correct. We use a two-tailed hypothesis test to determine whether the estimated value of a population
parameter is different from the hypothesized value. In a two-tailed test, we reject the null in favor of the alternative if
the evidence indicates that the population parameter is either smaller or larger than the value of the parameter under
H0. The null hypothesis for this test will be H0 = 6.0. Section 2. LO.d.
Question
6
Q-Code: L1-QM-HYPT-013
LOS e
Section 2
What type of consideration is an investor‟s tolerance for risk and financial position in hypothesis testing?
A) Investment or economic decision.
B) Statistical decision.
C) Both statistical and economic decision.
Answer
A) Investment or economic decision.
Explanation A is correct. A statistical decision simply consists of rejecting or not rejecting the null hypothesis. Whereas, an
economic decision takes into consideration all economic issues relevant to the decision, such as transaction costs,
risk tolerance and the impact on the existing portfolio. Investor’s risk tolerance is an investment or economic
decision, and not a statistical decision. Section 2. LO.e.
Question
7
Q-Code: L1-QM-HYPT-016
LOS f
Section 2
A researcher conducted a one-tailed test with the null hypothesis that the mean of a distribution is greater than 2.
The p-value came out to be 0.0475. If the researcher decides to use a 5% level of significance, the best conclusion is
to:
A) fail to reject the null hypothesis.
B) reject the null hypothesis.
C) decrease the level of significance to 4.75%.
Answer
B) reject the null hypothesis.
Explanation B is correct. If the p-value is lower than our specified level of significance, we reject the null hypothesis. Because the
p-value (0.0475) is lower than the stated level of significance (0.05), we will reject the null hypothesis. Section 2. LO.f
Question
8
Q-Code: L1-QM-HYPT-011
LOS d
Section 2
Assume that the population mean is μ, sample mean is x̅ and sx̅ is the standard error of the sample mean. Which of
the following is a condition for rejecting the null hypothesis at the 95 percent confidence interval ?
A) (X̄ - μ0) / sx̅ > 1.96
B) (X̄ - μ0) > 1.96
C) μ0 - X̄ / sx̅ > 1.96
Answer
A) (X̄ - μ0) / sx̅ > 1.96
Explanation A is correct.
At 5% level of significance or 95% confidence interval, we calculate the z-values that correspond to 0.05/2 = 0.025
level of significance. These are +1.96 and -1.96.
We reject the null hypothesis if we find that the test statistic ((X̄ - μ 0) / s x̅) is less than -1.96 or greater than +1.96.
Section 2. LO.d.
Question
9
Q-Code: L1-QM-HYPT-002
Section 2
Marco Vitaly is a researcher and wants to test whether a particular parameter is larger than a specific value. In this
case, the null and alternative hypothesis would be best defined as:
A) H${}_{0}$: $theta$ = $theta
0$versusH$a$
: $θ$$≠$$θ
{}_{0}$.
B) H${}_{0}$: $theta$ $mathrm{le}$ $theta
0$versusH$a$
: $θ$$ > $$θ
{}_{0}$.
C) H${}_{0}$: $theta$ $mathrm{ge}$ $theta
0$versusH$a$
{}_{0}$.
Answer
LOS a
B) H${}_{0}$: $theta$ $mathrm{le}$ $theta
: $θ$$ < $$θ
0$versusH$a$
: $θ$$ > $$θ
{}_{0}$.
Explanation B is correct. A positive "hoped for'' condition means that we will only reject the null (and accept the alternative) if the
evidence indicates that the population parameter is greater than $theta
0$. Thus, H$0$
: $θ$$≤$$θ
{}_{0}$ versus Ha: $theta$ $>$ $theta$${}_{0}$ is the correct statement of the null and alternative hypotheses.
Section 2. LO.a.
Question
10
Q-Code: L1-QM-HYPT-004
LOS b
Section 2
Which of the following statements requires a two-tailed test ?
A) H${}_{0}$: µ $mathrm{le}$ 0 versus H${}_{a}$: µ $>$ 0.
B) H${}_{0}$: µ = 0 versus H${}_{a}$: µ $mathrm{neq}$ 0.
C) H${}_{0}$: µ $mathrm{ge}$ 0 versus H${}_{a}$: µ $<$ 0.
Answer
B) H${}_{0}$: µ = 0 versus H${}_{a}$: µ $mathrm{neq}$ 0.
Explanation B is correct. A two-tailed test for the population mean is structured as: Ho : µ = 0 versus Ha : µ ≠ 0. Section 2. In a
two-tailed test, we reject the null in favor of the alternative if the evidence indicates that the population parameter is
either smaller or larger than the value of the parameter under H0. LO.b.
Question
11
Q-Code: L1-QM-HYPT-006
LOS c
Section 2
In order to calculate the test statistic, the difference between the sample statistic and the value of the population
parameter under Ho is most likely divided by:
A) appropriate value from the t-distribution.
B) sample standard deviation.
C) standard error of the sample statistic.
Answer
C) standard error of the sample statistic.
Explanation C is correct. A test statistic is defined as the difference between the sample statistic and the value of the population
parameter under Ho divided by the standard error of the sample statistic.
test statistic = (sample statistic - value of the parameter under H0) / standard error of the sample statistic
Section 2. LO.c.
Question
12
Q-Code: L1-QM-HYPT-008
LOS c
Section 2
The results of an experiment are statistically significant when:
A) the null hypothesis is rejected.
B) the null hypothesis is not rejected.
C) the level of significance is altered.
Answer
A) the null hypothesis is rejected.
Explanation A is correct. The results of an experiment are statistically significant when the null hypothesis is rejected. Section 2.
LO.c.
Question
13
Q-Code: L1-QM-HYPT-010
LOS d
Section 2
The mean annual return is 8 percent and the standard deviation is 6.4 percent for a sample containing 25 sectors. A
fund manager is testing whether the mean annual return is less than 9 percent. The critical value is -1.96. What is
the most likely conclusion from this test ?
A) Reject the null hypothesis.
B) Do not reject the null hypothesis.
C) Additional information is required to decide.
Answer
B) Do not reject the null hypothesis.
Explanation B is correct.
Step 1: State the hypothesis
H0: µ ≥ 9%
Ha : µ < 9%
Step 2: Calculate the test statistic
Step 3: Calculate the critical value
This is a one-tailed test and we will be looking at the left tail. The critical value given, which is -1.96
Step 4: Decision
Since the test statistic is less negative (lower absolute value) than the critical value (1.96), the null hypothesis is not
rejected. Section 2. LO.d.
Question
14
Q-Code: L1-QM-HYPT-012
LOS e
Section 2
Rejecting or not rejecting the null hypothesis is a:
A) Statistical decision.
B) Economic decision.
C) Both statistical and economic decision.
Answer
A) Statistical decision.
Explanation A is correct. A statistical decision simply consists of rejecting or not rejecting the null hypothesis. Whereas, an
economic decision takes into consideration all economic issues relevant to the decision, such as transaction costs,
risk tolerance and the impact on the existing portfolio. Section 2. LO.e.
Question
15
Q-Code: L1-QM-HYPT-014
LOS f
Section 2
Which of the following statements regarding the p-value is most likely to be correct?
A) The p-value is the smallest level of significance at which the null hypothesis can be rejected.
B) The p-value is the smallest level of significance at which the null hypothesis can be accepted.
C) The p-value is the largest level of significance at which the null hypothesis can be rejected.
Answer
A) The p-value is the smallest level of significance at which the null hypothesis can be rejected.
Explanation A is correct. The p-value is defined as the smallest level of significance at which the null hypothesis can be rejected. It
can be used in the hypothesis testing framework as an alternative to using rejection points.
If the p-value is lower than our specified level of significance, we reject the null hypothesis.
If the p-value is greater than our specified level of significance, we do not reject the null hypothesis.
For example, if the p-value of a test is 4%, then the hypothesis can be rejected at the 5% level of significance, but
not at the 1% level of significance. Section 2. LO.f.
Question
16
Q-Code: L1-QM-HYPT-015
LOS f
Section 2
A researcher formulates a null hypothesis that the mean of a distribution is equal to 20. He obtains a p-value of
0.018. Using a 5% level of significance, the best conclusion is to:
A) reject the null hypothesis.
B) accept the null hypothesis.
C) decrease the level of significance.
Answer
A) reject the null hypothesis.
Explanation A is correct. If the p-value is lower than our specified level of significance, we reject the null hypothesis. As the pvalue of 0.018 is less than the stated level of significance (0.05), we reject the null hypothesis. Section 2. LO.f.
Question
17
Q-Code: L1-QM-HYPT-017
LOS f
Section 2
A researcher is using the p-value test for conducting hypothesis testing. He is most likely to reject the null
hypothesis when the p-value of the test statistic:
A) exceeds a specified level of significance.
B) falls below a specified level of significance.
C) is negative.
Answer
B) falls below a specified level of significance.
Explanation B is correct. If the p-value is less than the specified level of significance, the null hypothesis is rejected. If the p-value
is greater than our specified level of significance, we do not reject the null hypothesis. For example, if the p-value of a
test is 4%, then the hypothesis can be rejected at the 5% level of significance, but not at the 1% level of significance.
Section 2. LO.f.
Question
18
Q-Code: L1-QM-HYPT-018
LOS f
Section 2
A researcher conducts a two-tailed t-test test with a null hypothesis that the population mean differs from zero. If
the p-value is 0.089 and he is using a significance level of 5%, the most appropriate conclusion is:
A) do not reject the null hypothesis.
B) reject the null hypothesis.
C) the chosen significance level is too high.
Answer
A) do not reject the null hypothesis.
Explanation A is correct. The p-value is the smallest level of significance at which the null hypothesis can be rejected. If the pvalue is less than the specified level of significance, the null hypothesis is rejected. If the p-value is greater than our
specified level of significance, we do not reject the null hypothesis. In this case, the given p-value is greater than the
given level of significance. Hence, we cannot reject the null hypothesis. Note that we simply compare the given pvalue with the level of significance. Even though this is a two-tailed test we do not divide the p-value by 2. Section 2.
LO.f
Question
19
Q-Code: L1-QM-HYPT-020
LOS g
Section 3
Orlando Bloom is analyzing a portfolio’s performance for the past 15 years. The mean return for the portfolio is
10.25% with a sample standard deviation of 12.00%. Bloom wants to test the claim that the mean return is less than
12.50%. The null hypothesis is that the mean return is greater than or equal to 12.50%. If the critical value for this
test is -2, which of the following is most likely the test statistic and the decision of this test?
Option
Test Statistic
Decision
A.
-0.726
Reject Ho
B.
-0.726
Do not reject Ho
C.
-0.5422
Do not reject Ho
A) Option A in the above table.
B) Option B in the above table.
C) Option C in the above table.
Answer
B) Option B in the above table.
Explanation B is correct.
Since the absolute value of -0.726 is less than the absolute value of -2, we cannot reject the null hypothesis. Section
2 and 3. LO.g.
Question
20
Q-Code: L1-QM-HYPT-022
LOS g
Section 3
Peter is studying the earnings per share of 32 companies in an industry. He plans to use the t-test for hypothesis
testing. The degrees of freedom Peter will use for defining the critical region is closest to:
A) 30.
B) 31.
C) 32.
Answer
B) 31.
Explanation B is correct. In a t-test, the degree of freedom is sample size (n) - 1. Therefore, it will be 32 – 1 = 31 in this case.
Section 3. LO.g.
Question
21
Q-Code: L1-QM-HYPT-019
LOS g
Section 3
Which of the following statistic is most likely to be used for the mean of a non-normal distribution with unknown
variance and a small sample size?
A) z test statistic.
B) t test statistic.
C) There is no test statistic for such a scenario.
Answer
C) There is no test statistic for such a scenario.
Explanation C is correct. The statistic for a small sample size of a non-normal distribution with unknown variance is not available.
The z-test statistic is used for a large sample size of a non-normal distribution with known variance while t-test
statistic is used for a large sample size of a non-normal distribution with unknown variance.
Refer to the table below:
Sampling from
Normal
distribution
Small sample
size (n<30)
Variance known
Variance
unknown
Non –normal
distribution
Variance known
Variance
unknown
Large
sample size
(n≥30)
z
z
t
t (or z)
NA
z
NA
t (or z)
Section 3. LO.g.
Question
22
Q-Code: L1-QM-HYPT-024
LOS i
Section 3
The table below shows the return data for samples which have been pooled from two normally distributed
populations with equal variance.
Sample #
Sample size
Annual returns
1
60
15.8%
2
112
12.5%
The standard deviation of the pooled sample, s, is 256.68. Which of the following is the correct test statistic to test
for the differences between means?
A) 0.0006.
B) 0.0008.
C) 0.0011.
Answer
B) 0.0008.
Explanation B is correct.
When we can assume that the two populations are normally distributed and that the unknown population variances
are unequal, an approximate t-test based on independent random samples is given by:
In this formula, we use the tables of the t-distribution using the ‘modified’ degrees of freedom. The ‘modified’
degrees of freedom are calculated using the following formula:
Section 3.1. L i.
Question
23
Q-Code: L1-QM-HYPT-023
LOS h
Section 3
From two normally distributed populations, independent samples were drawn and followingobservations were made:
Sample A: The sample size of 20 observations had a sample mean of 63.
Sample B: The sample size of 14 observations had a sample mean of 58.
Standard deviations of sample A and sample B were equal. The pooled estimate of common variance was equal
to 565.03.
A researcher devised the hypothesis that the two sample means are equal. In order to test this hypothesis, the ttest statistic to be used is closest to:
A) 0.21.
B) 0.35.
C) 0.60.
Answer
C) 0.60.
Explanation C is correct. When we assume that the two populations are normally distributed and that the unknown population
variances are equal, the t-test based on independent random samples is given by:
The term is known as the pooled estimator of the common variance. It is calculated by the following formula:
The number of degrees of freedom is n1 + n2 – 2.
= 0.604.. Section 3.1. LO.h.
Question
24
Q-Code: L1-QM-HYPT-027
LOS i
Section 3
Which of the following is true for a paired comparison test?
A) The samples are independent.
B) The samples are dependent.
C) The test conducted is a test concerning differences between mean and not mean differences.
Answer
B) The samples are dependent.
Explanation B is correct. A paired comparison test is conducted for mean differences and the samples are dependent. For
example, suppose you want to conduct tests on the mean monthly return on Toyota stock and mean monthly
return on Honda stock. These two samples are believed to be dependent, as they are impacted by the same
economic factors. Section 3.2. LO.i.
Question
Q-Code: L1-QM-HYPT-026
LOS i
Section 3
25
An analyst collects the following data related to paired observations for Sample A and Sample B. Assume that both
samples are drawn from normally distributed populations and that the population variances are not known:
Paired
Observation
Sample A
Value
Sample B
Value
1
12
5
2
18
15
3
4
1
4
-6
-9
5
-5
4
The t-statistic to test the hypothesis that the mean difference is equal to zero is closest to:
A) 0.23.
B) 0.27.
C) 0.52.
Answer
Explanation
C) 0.52.
Paired
Sample
Sample
Differences
Differences Minus
the Mean
Observation
A
Value
B
Value
1
12
5
7
(7-1.4)2=31.36
2
18
15
3
(3-1.4)2=2.56
3
4
1
3
(3-1.4)2=2.56
4
-6
-9
3
(3-1.4)2=2.56
5
-5
4
-9
(-9-1.4)2=108.16
Sum = 7
Mean = 1.4
Sum of squared
differences = 147.2
Sample
variance:
147.2 / 4 = 36.8
Difference, Then
Squared
Standard
error:
t-Statistic:
1.4 - 0 / 2.712932 =
0.51605
Section 3.2. LO.i.
Question
26
Q-Code: L1-QM-HYPT-028
The table below shows the annual return summary for KSE-50 and KSE-100 portfolios.
Portfolio
Mean
Return
Standard
Deviation
KSE – 50
19.25%
20.05%
KSE – 100
15.98%
17.11%
Difference
3.27%
5.48%
LOS i
Section 3
The null hypothesis for the test conducted is Ho: µd = 0. The sample size is 64.
Which of the following most likely represent the test conducted and the value of the test statistic?
A) A chi square test with t statistic = 4.77.
B) A paired comparison test with t statistic = 5.27.
C) A paired comparison test with t statistic = 4.77.
Answer
C) A paired comparison test with t statistic = 4.77.
Explanation C is correct. Since the test concerns mean differences, it is a paired comparisons test.
t = (đ - μd0) / s đ = (3.27 - 0) / (5.48 / √64) = 4.77.
Section 3.2. LO.i.
Question
27
Q-Code: L1-QM-HYPT-029
LOS i
Section 3
A hypothesis test is to be conducted in order to test the differences between means. Which of the following will least
likely be used as a null hypothesis for this test?
A) H${}_{o}$: µ${}_{1}$ + µ${}_{2}$ = 0.
B) H${}_{o}$: µ${}_{1}$ - µ${}_{2}$ = 0.
C) H${}_{o}$: µ${}_{1 } leq $µ${}_{2}$.
Answer
A) H${}_{o}$: µ${}_{1}$ + µ${}_{2}$ = 0.
Explanation A is correct. Since we want to test whether the two means were equal or different, we define the hypotheses as:
H0: µ1 - µ2 = 0 or
H0: µ1 < µ2.
The incorrect null hypothesis is Ho : µ1 + µ2 = 0.
Section 3.2. LO.i.
Question
28
Q-Code: L1-QM-HYPT-025
LOS i
Section 3
You want to test a hypothesis related to the mean of the difference between Honda and Toyota stock returns at the
1% level of significance. Hence you calculate the difference between Honda and Toyota returns over the last 25
months. For this data, the sample mean difference is 3.25 and the sample standard deviation is 6.70. The critical
value for this test is 2.8. The most appropriate conclusion is to:
A) reject the hypothesis that the mean difference equals zero as the computed test statistic exceeds 2.8.
B) accept the hypothesis that the mean difference equals zero as the computed test statistic exceeds 2.8.
C) accept the hypothesis that the mean difference equals zero as the computed test statistic is less than 2.8.
Answer
C) accept the hypothesis that the mean difference equals zero as the computed test statistic is less than 2.8.
Explanation C is correct. Paired observations are observations that are dependent because they have something in common. In
such situations, we conduct a t-test that is based on data arranged in paired observations.
As 2.4 < critical value of 2.8, we do not reject the null hypothesis that the mean difference is zero. Section 3.3. LO.i.
Question
29
Q-Code: L1-QM-HYPT-021
LOS g
Section 4
The test statistic for hypothesis test of a single mean where the population sample has unknown variance is most
likely:
A) (X̄ - μ) / (s/√n)
B) (n-1)(s2) / σ2
C) s1 2 / s22
Answer
A) (X̄ - μ) / (s/√n)
Explanation A is correct. The test statistic for the hypothesis test of a single mean where the population sample has unknown
variance is (X̄ - μ) / (s/√n). This would require a t-test.
(n-1)(s 2) / σ02 is chi-square test statistic (χ²)
s1
2
/ s 22 is F-statistic.
Section 4. LO.g.
Question
30
Q-Code: L1-QM-HYPT-033
LOS j
Section 4
Which test should be used for hypothesis related to a single population variance?
A) A chi-square test with degrees of freedom, n.
B) A chi-square test with degrees of freedom, n-1.
C) An F-test with degrees of freedom, n-1.
Answer
B) A chi-square test with degrees of freedom, n-1.
Explanation B is correct. In tests concerning the variance of a single normally distributed population, we use the chi-square test
statistic, denoted by χ2. The chi-square distribution is asymmetrical and like the t-distribution, is a family of
distributions. It uses (n – 1) degrees of freedom. The test statistic is
χ2 = (n -1)(s 2) / σ02
where:
n = sample size
s = sample variance
Option C is incorrect because the F-test is used in order to test the equality or inequality of two variances. The F-test
is the ratio of sample variances.
The assumptions for a F-test to be valid are:
The samples must be independent.
The populations from which the samples are taken are normally distributed.
Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees
of freedom.
The formula for the test statistic of the F-test is:
F = s 12 / s 22
where:
s 12 = the sample variance of the first population with n observations
s 22 = the sample variance of the second population with n observations
A convention is to put the larger sample variance in the numerator and the smaller sample variance in the
denominator.
df1 = n1 – 1 numerator degrees of freedom
df2 = n2 – 1 denominator degrees of freedom
Section 4.1. LO.j.
Question
31
Q-Code: L1-QM-HYPT-030
LOS j
Section 4
A researcher drew two samples from two normally distributed populations. The mean and standard deviation of the
first sample were 4 and 48 respectively. The mean and standard deviation of the second sample were 6 and 52
respectively. The number of observations in the first sample was 30 and second sample was 32. Given a null
hypothesis of ${sigma }^2_1= {sigma }^2_2$ versus an alternate hypothesis of ${sigma }^2_1neq {sigma
}^2_2 $, which of the following is most likely to be the test statistic?
A) 0.235.
B) 0.852.
C) 1.170.
C) 1.170.
Answer
C) 1.170.
Explanation C is correct. The test that compares the variances using two independent samples from two different populations
makes use of the F-distributed t-statistic:
The formula for the test statistic of the F-test is:
A convention is to put the larger sample variance in the numerator and the smaller sample variance in the
denominator.
df1 = n1 – 1 numerator degrees of freedom
df2 = n2 – 1 denominator degrees of freedom
The test statistic is then compared with the critical values found using the two degrees of freedom and the F-tables.
The smaller variance is the denominator, thus:
522 / 482 = 1.17
Option B is incorrect because it uses 482 / 522 = 0.852.
Section 4.2. LO.j.
Question
32
Q-Code: L1-QM-HYPT-032
LOS j
Section 4
For an F-test specified as
$frac{{mathrm{s}}^{mathrm{2}}_{mathrm{1}}}{{mathrm{s}}^{mathrm{2}}_{mathrm{2}}}$, which of the
following is used as the actual test statistic?
A) ${mathrm{s}}_{mathrm{1}}
$shouldbegreaterthan$s 2
{}_{. }$
B) ${}_{ }
s 1$shouldbelessthan$s 2
{}_{.}$
C) It does not matter whether ${mathrm{s}}_{mathrm{1}}
$isgreaterorlessthan$s 2
{}_{.}$
Answer
A) ${mathrm{s}}_{mathrm{1}}
$shouldbegreaterthan$s 2
{}_{. }$
Explanation A is correct. A convention is to put the larger sample variance in the numerator and the smaller sample variance in
the denominator, which is only possible if option A is true. Section 4.2. LO.j.
Question
33
Q-Code: L1-QM-HYPT-031
LOS j
Section 4
The null hypothesis ${mathrm{H}}_{mathrm{o}}mathrm{:
}{mathrm{sigma }}^{mathrm{2}}_{mathrm{1}}mathrm{=}{mathrm{sigma }}^{mathrm{2}}_{mathrm{2}}$
most likely tests:
A) the mean differences.
B) a single variance.
C) the equality of two variances.
Answer
C) the equality of two variances.
Explanation C is correct. The test concerns the equality of two variances. It is known as the F-test.
When working with F-tests, there are three hypotheses that can be formulated:
1. H0: σ12 = σ22 versus Ha : σ12 ≠ σ22. This is used when we believe the two population variances are not equal.
2. H0: σ12 ≤ σ22 versus Ha : σ12 > σ22. This is used when we believe the variance of the first population is greater
than the variance of the second population.
3. H0: σ12 ≥ σ22 versus Ha : σ12 < σ22. This is used when we believe the variance of the first population is less than
the variance of the second population.
The term σ12 represents the population variance of the first population and σ22 represents the population variance of
the second population.
The formula for the test statistic of the F-test is:
F = s 12 / s 22
where:
s 12 = the sample variance of the first population with n observations
s 22= the sample variance of the second population with n observations
A convention is to put the larger sample variance in the numerator and the smaller sample variance in the
denominator.
df1 = n1 – 1 numerator degrees of freedom
df2 = n2 – 1 denominator degrees of freedom
The test statistic is then compared with the critical values found using the two degrees of freedom and the Ftables. Section 4.2. LO.j.
For mean differences we use the following: H 0: µd = µd0 versus Ha : µd ≠ µd0
µd stands for the population mean difference and µd0 stands for the hypothesized value for the population mean
difference.
For single mean difference, we use the following:
The null and alternative hypotheses for this example will be:
H0: µ = 0 versus Ha: µ ≠ 0
Question
34
Q-Code: L1-QM-HYPT-048
LOS k
Section 4
The sample correlation between the current account (% of GDP) and fiscal balance (% of GDP) of a Country is
0.6345 for the period from June 2005 through June 2015. Assuming 0.05 level of significance, which of the following
is most likely correct?
A) The test statistic is 8.96, so we can reject the null hypothesis.
B) The test statistic is 8.92, so we can reject the null hypothesis.
C) The test statistic is 8.92, so we cannot reject the null hypothesis.
Answer
B) The test statistic is 8.92, so we can reject the null hypothesis.
Explanation B is correct. With 120 months from June 2005 through June 2015, we use the following statistic to test the null
hypothesis, H0, that the true correlation in the population is 0, against the alternative hypothesis, Ha , that the
correlation in the population is different from 0:
t = [0.6345(120 - 2)½] / [1 - 0.63452] = 8.92
At the 0.05 significance level, the critical level for this test statistic is 1.984 (n = 120, degrees of freedom = 118).
When the test statistic is either larger than 1.984 or smaller than 1.984, we can reject the hypothesis that the
correlation in the population is 0. The test statistic is 8.92, so we can reject the null hypothesis. Section 4.3. LO. k.
Question
35
Q-Code: L1-QM-HYPT-035
LOS k
An investment analyst will least likely use a non-parametric test in which of the following situations?
Section 5
A) When the data does not meet distributional assumptions.
B) When the data provided is given in ranks.
C) When the hypothesis being addressed concerns a parameter.
Answer
C) When the hypothesis being addressed concerns a parameter.
Explanation C is correct. We use nonparametric procedures in three situations:
Data does not meet distributional assumptions.
Data is given in ranks. (Example: relative size of the company and use of derivatives.)
The hypothesis does not concern a parameter. (Example: Is a sample random or not?)
Section 5. LO.k.
Question
36
Q-Code: L1-QM-HYPT-034
LOS k
Section 5
A test that makes minimal assumptions about the population from which the sample comes is known as a:
A) paired comparisons test.
B) parametric test.
C) nonparametric test.
Answer
C) nonparametric test.
Explanation C is correct. Nonparametric tests are not concerned with a parameter and/or make few assumptions about the
population from which the sample are drawn. We use nonparametric procedures in three situations:
Data does not meet distributional assumptions.
Data is given in ranks. (Example: relative size of the company and use of derivatives.)
The hypothesis does not concern a parameter. (Example: Is a sample random or not?)
The Spearman rank correlation coefficient test is a popular nonparametric test. The coefficient is calculated based on
the ranks of two variables within their respective samples.
A parametric test is a hypothesis test concerning a parameter or a hypothesis test based on specific distributional
assumptions.
The paired comparison test is a hypothesis test on differences between the means of two populations when two
samples are believed to be dependent.
Section 5. LO.k.