UNIVERSITY OF MELBOURNE
DEPARTMENT OF ECONOMICS
SEMESTER 1 ASSESSMENT, 2020
ECOM30001: Basic Econometrics
Time Allowed: THREE Hours
Reading Time: 30 minutes
This examination paper contributes a minimum of 60 percent to the assessment in ECOM30001.
This examination consists of nine (9) questions in total.
You must answer ALL nine questions.
The maximum number of marks for this examination is 80 marks. The marks allocated to each
question are as follows:
Question 1 5 marks
Question 2 5 marks
Question 3 5 marks
Question 4 5 marks
Question 5 20 marks
Question 6 17 marks
Question 7 3 marks
Question 8 6 marks
Question 9 14 marks
This exam has 23 pages.
The examination contains a formula sheet starting on page 18.
The examination contains critical values for a number of distributions starting on page 21.
Page 1 of 23
Question 1 [5 marks]
Suppose you are investigating the impact of some policy change for some outcome yit. You
have selected two groups of individuals for your analysis—a control group and a treatment
group. You also have available two observations for these individuals—for a period before the
policy change and a period after the policy change. Consider the following econometric model
for a ‘difference-in-difference’ research design for outcome yit:
yit = α0 + α1 treati + α2 aftert + β1 {treati ∗ aftert}+ εit
where:
treati =
{
1 if individual i is in the treatment group
0 if individual i is in the control group
and:
aftert =
{
1 if period is after treatment
0 if period is before treatment
a) Assuming the following restriction is satisfied, what is the interpretation of the population
parameter α1?
{E[εit|treati = 1, aftert = 0]− E[εit|treati = 0, aftert = 0]} = 0
b) Assuming the following restriction is satisfied, what is the interpretation of the population
parameter α2?
{E[εit|treati = 0, aftert = 1]− E[εit|treati = 0, aftert = 0]} = 0
c) Assuming the following restriction is satisfied, what is the interpretation of the population
parameter β1?
{E[εit|treati = 1, aftert = 1]− E[εit|treati = 1, aftert = 0]}
− {E[εit|treati = 0, aftert = 1]− E[εit|treati = 0, aftert = 0]} = 0
Page 2 of 23
Question 2 [5 marks]
Consider the following econometric model:
∆ uratet = β0 + β1 uratet−1 + β2 ∆ uratet−1 + β3 ∆ uratet−2
+ β4 ∆ uratet−3 + β5 ∆ uratet−4 + εt
where urate represents the quarterly unemployment rate. This econometric model was esti-
mated using the method of Ordinary Least Squares (OLS) for the period 1978:Q1 to 2019:Q4
and the results are presented in Figure 1.





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Figure 1: OLS Regression Results: Question 2
Outline how you would test whether the series for the quarterly unemployment is stationary
or not. Your answer should clearly state the null and alternative hypotheses, the test statistic
and its distribution. Using the results in Figure 1, what is the value of the Augmented Dickey-
Fuller test statistic? At the 5% level of significance, explain whether the sample evidence is
consistent with the null hypothesis. Based upon the estimation results presented in Figure 1,
the p-value associated with the Augmented Dickey-Fuller test is 0.03175.
Page 3 of 23
Question 3 [5 marks]
Consider the following demand function for the quantity demanded of product Qt as a function
of its market price Pt
lnQt = α0 + α1 lnPt + εDt
Consider also the following supply function for the quantity supplied of product Qt as a function
of its market price Pt
lnQt = β0 + β1 lnPt + β2Xt + εSt
with COV(Xt, εDt) = 0, COV(Xt, εSt) = 0, and COV(εDt, εSt) = 0.
Write out an expression for the reduced form for lnPt as a function of Xt. Does the demand
equation satisfy the necessary condition for identification? Briefly explain how you would test
that the demand function satisfies the necessary condition(s) for identification. Your answer
should clearly state the null and alternative hypotheses, the test statistic and its distribution.
Page 4 of 23
Question 4 [5 marks]
Let yi denote the number of cigarettes smoked per day by individual i. The (probability)
density of yi follows a Poisson distribution
f(yi|λi) = exp(−λi)λ
yi
i
yi!
i = 1, 2, . . . N
with E[yi] = λi and VAR[yi] = λi.
Assume the conditional expectation of the number of cigarettes smoked per day is given by:
E[cigsi|Xi] = exp
{
β0 + β1 ln cigpricei + β2educi + β3agei + β4age
2
i + β5resti
}
where:
yi = cigs = number of cigarettes smoked per day for individual i
cigprice = price paid per packet of cigarettes by indivudual i
educ = years of education of individual i
age = age in years of individual i
rest = 1 if individual i faces smoking restrictions in restaurants, 0 otherwise
a) What is meant by the term count data? What are the important characteristics of count
data?
b) What is the interpretation of the parameter β1?
c) What is the interpretation of the parameter β5?
Page 5 of 23
Question 5 [20 marks]
Consider the following econometric model relating years of education to the number of children
ever born to women in a developing country:
childreni = β0 + β1 educi + β2 agei + β3 age
2
i + β4 urbani + β5 age firsti + εi (1)
where:
children = total number of chidren ever born to mother i
educ = total years of education for mother i
age = current age, in years, of mother i
urban = 1 if mother i resides in an urban area, 0 otherwise
age first = age at first birth for mother i, in years
Using a sample of women aged 15-49, the econometric model (1) was estimating by the method
of Ordinary Least Squares (OLS) and the results are presented in Figure 2.







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Figure 2: Comparing the OLS and IV Regression Results for Model (1)
a) [3 marks] Do you think the condition COV(educ, ε) = 0 in model (1) is likely to be
satisfied? Clearly explain why or why not. Explain the consequences for the OLS esti-
mator if this condition is not satisfied. Clearly outline at least one possible reason why
this condition might not be satisfied.
b) [3 marks] Consider the following indicator variable:
firsthalf = 1 if mother was born in the first six months of the year, 0 otherwise
Page 6 of 23
Some researchers have suggested using the variable firsthalf as an instrumental variable
for the variable educ. Why? Individuals born in different months of the year start school
at different ages, while compulsory schooling laws generally require students to remain in
schools until a specific birthday. Effectively, the interaction of school-entry requirements
and compulsory schooling laws compel students born in certain months to attend school
for a longer period than students born in other months. In the current application, this
interaction implies mothers born in the first half of the year will start school at an older
age than mothers born in the second half of the year. With a fixed age for minimum
compulsory schooling, mothers born in the first half of the year will have, on average,
less years of completed schooling.
Clearly explain the two conditions that must be satisfied for the variable firsthalf to
be a valid instrumental variable. Do you think these two conditions are likely to be
satisfied? Why or why not?
c) [3 marks] Clearly explain what is meant by the Weak Instruments problem. Explain
the consequences for statistical inference using the Instrumental Variable estimator (IV)
with weak instruments.
d) [2 marks] Consider the following reduced form:
educi = pi0 + pi1 firsthalfi + pi2 age + pi3 age
2
i + pi4 urbani + pi5 age firsti + εi (2)
The results from estimating this reduced form by OLS are reported in Figure 3. Based
upon these results, explain whether the variable firsthalf is an adequate instrumental
variable for educ.







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Figure 3: OLS Estimation Results for the Reduced Form Model (2)
e) [2 marks] The estimation results from estimating model (1) by the method of Instru-
mental Variables (IV) are reported in Figure 2. Compare and contrast the estimate of β1
obtained using OLS with that obtained using the method of IV. In your answer comment
Page 7 of 23
on both the magnitude of the estimate and the statistical significance of the estimate of
β1.
f) [3 marks] Provided firsthalf is a valid instrument, the IV estimator for β1 in model (1)
will provide a consistent estimator for the causal effect of education upon female fertility.
Comparing your estimates for β1 using the IV estimator, with those obtained using the
OLS estimator, provide and explain a possible source generating COV(educ, ε) 6= 0 in
model (1) when using the OLS estimator.
g) [2 marks] Using model (1), write down an expression for the marginal (partial) effect of
age on the number of children ever born. Consider the results from estimating model (1)
by the method of Instrumental Variables (IV). Is this estimated marginal effect positive
for all women in the sample. Clearly explain why or why not?
h) [2 marks] Figure 4 provides a plot of the predicted values for children from model (1)
estimated using the Instrumental Variables Estimator. Based upon this Figure, do you
think that the linear model (1) is the correct specification for the number of children ever
born. Clearly explain why or why not?
0
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40
60
80
−
2
−
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Predicted Number of Children
Co
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Figure 4: Plot of Predicted Values for Model (1), IV Estimator
Page 8 of 23
Question 6 [17 marks]
Consider the following econometric model for the hourly wages for a group of workers:
ln wageit = β0 + β1 educi + β2 experit + β3 exper
2
it + β4 unionit + β5 cityit + εit (3)
where lnX denotes the natural logarithm of variable X and:
wage = hourly wages of individual i in time t
educ = years of completed education of individual i
exper = years of labour market experience of individual i in time t
union = 1 if individual i is a member of a union in time t, 0 otherwise
city = 1 if individual i resides in a large city in time t, 0 otherwise
Economic theory suggests the following restrictions on the population parameters:
β1 > 0, β2 > 0, β3 < 0, β4 > 0, β5 > 0
Suppose you have a dataset with observations on 545 workers who have worked continuously
over eight (8) years, providing a total of 4,360 total observations.
a) [2 marks] Suppose you estimate model (3) by Ordinary Least Squares (OLS). Do you
think that the standard errors are valid? Clearly explain why or why not?
b) Consider the following alternative specification for the hourly wages for a group of work-
ers:
ln wageit = β0 + β2 experit + β3 exper
2
it + β4 unionit + β5 cityit + υi + εit (4)
where υi represents an unobserved time-invariant random variable.
i) [2 marks] Suppose you estimate the econometric model (4) using the Fixed Ef-
fects (FE) estimator. Clearly explain why your estimated model cannot include the
variable educ.
ii) [4 marks] Clearly explain the assumption about the relationship between υi and
each of the X variables that is imposed when estimating model (4) using the Fixed
Effects (within) estimator. Clearly explain, and provide an example, whether you
think that this is a realistic assumption.
iii) [3 marks] The estimation results for model (3) using the OLS estimator and the
estimation results for model (4) using the Fixed Effects (FE) estimator are presented
in Figure 5. Compare and contrast the estimates for model (3) to those for model
(4) estimated using the Fixed Effects (FE) estimator. Based on this comparison
comment on the likely sign of any omitted variable bias in model (3).
iv) [4 marks] Clearly outline the important differences between the Random Effects
(RE) estimator and the Fixed Effects (FE) estimator. Your answer should clearly
explain the variation in the data that is used to identify the parameters of interest.
v) [2 marks] Since the available data span a period of eight years you decide that it
might be sensible to include a linear trend in the model. Consider the following
alternative specification:
ln wageit = β0 + β2 experit + β3 exper
2
it + β4 unionit + β5 cityit + β6 t+ υi + εit (5)
where:
t = a linear trend, t = 1, 2, 3, . . . 8
Suppose you attempt to estimate this alternative model (5) using the Fixed Effects
(FE) estimator. Clearly explain why the parameters β2 and β6 are not separately
identified.
Page 9 of 23





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Page 10 of 23
Question 7 [3 marks]
Consider the following Correlated Random Effects (CRE) model:
ln wageit = β0 + β2 experit + β3 exper
2
it + β4 unionit + β5 cityit
+ β6 experi + β7 exper
2
i + β8 unioni + β9 cityi + ηi + εit (6)
where lnX denotes the natural logarithm of variable X and ηi represents an unobserved
time-invariant random variable which satisfies the restriction COV(ηi,Xit) = 0 for each
of the explanatory variables.
wage = hourly wages of individual i in time t
educ = years of completed education of individual i
exper = years of labour market experience of individual i in time t
union = 1 if individual i is a member of a union in time t, 0 otherwise
city = 1 if individual i resides in a large city in time t, 0 otherwise
and:
experi = sample mean of exper for individual i over time
exper2i = sample mean of exper for individual i over time
unioni = sample mean of union for individual i over time
cityi = sample mean of city for individual i over time
Economic theory suggests the following restrictions on the population parameters:
β1 > 0, β2 > 0, β3 < 0, β4 > 0, β5 > 0
Suppose you have a dataset with observations on 545 workers who have worked continu-
ously over eight (8) years, providing a total of 4,360 total observations.
The estimation results for the CRE model (6) are presented in Figure 6. Using the results
in either Figure 7 or Figure 8 test the hypothesis that the Random Effects (RE) estimator
is the most appropriate model, at the 5% level of significance. Your answer should clearly
state the null and alternative hypotheses, the distribution of the test statistic, and your
decision.
Page 11 of 23





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Linear hypothesis test
Hypothesis:
exper = 0
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metro = 0
Model 1: restricted model
Model 2: lnwage ~ educ + exper + expersq + union + metro + mexper + mexper
sq +
munion + mmetro
Note: Coefficient covariance matrix supplied.
Res.Df Df Chisq Pr(>Chisq)
1 4354
2 4350 4 430.19 < 0.00000000000000022 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Figure 7: H0 : β2 = β3 = β4 = β5 = 0 in Correlated Random Effects (CRE) Model (6)
Linear hypothesis test
Hypothesis:
mexper = 0
mexpersq = 0
munion = 0
mmetro = 0
Model 1: restricted model
Model 2: lnwage ~ educ + exper + expersq + union + metro + mexper + mexper
sq +
munion + mmetro
Note: Coefficient covariance matrix supplied.
Res.Df Df Chisq Pr(>Chisq)
1 4354
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---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Figure 8: H0 : β6 = β7 = β8 = β9 = 0 in Correlated Random Effects (CRE) Model (6)
Page 12 of 23
Question 8 [6 marks]
A chain of national supermarkets is offering a new line of organic, environmentally friendly
products. In order to focus their marketing strategy, the management wants to determine the
characteristics of the customers most likely to purchase these products.
The supermarket chain has a customer loyalty program. As an initial buyer incentive plan, the
supermarket has provided coupons for these organic products to all of their loyalty program
participants and collected some data that includes whether these customers purchased any of
the organic products.
Consider the following econometric model:
buy∗i = β0 + β1 agei + β2 age
2
i + β3 tenurei + β4 metroi
+ β5 loyalty1i + β6 loyalty2i + β7 loyalty3i
+ β8 region1i + β9 region2i + β10 region3i + β11 region4i + εi (7)
where εi|Xi ∼ (0, σ2ε) and:
buy∗ = a latent variable that determines the decision to purchase organic products
age = Customer age, in years
tenure = length of time member of loyalty program
metro = 1 if customer lives in a metro area, 0 otherwise
regionj = 1 if customer lives in region j, 0 otherwise, where j = 1, 2, 3, 4, or 5
loyaltyj = 1 if customer has loyalty status j, 0 otherwise, where j = 1, 2, 3, or 4
The omitted categories for the dummy variables are region5, and loyalty4.
a) [1 mark] Provide a brief interpretation of the latent variable buy∗i
b) Suppose you have a dataset of 3,171 observations which, in addition to the required data,
also provides the following indicator variable:
buy = 1 if individual i puchases any organic products, 0 otherwise
This suggests the following econometric model:
buyi = β0 + β1 agei + β2 age
2
i + β3 tenurei + β4 metroi
+ β5 loyalty1i + β6 loyalty2i + β7 loyalty3i
+ β8 region1i + β9 region2i + β10 region3i + β11 region4i + εi (8)
where:
buyi =
{
1 if buy∗i ≥ 0
0 if buy∗i < 0
Consider estimating model (8) by Ordinary Least Squares (OLS) using buy as the de-
pendent variable. The results are provided in Figure 9.
i) [2 marks] What is the interpretation of the estimate of the parameter β3? What
is the interpretation of the estimate of parameter β4?
ii) [3 marks] Outline an issue that might arise with the predicted values from the
estimation of model (8) by the method of Ordinary Least Squares? Do you think
that the standard errors are valid? Clearly explain why or why not?
Page 13 of 23









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Page 14 of 23
Question 9 [14 marks]
A chain of national supermarkets is offering a new line of organic, environmentally friendly
products. In order to focus their marketing strategy, the management wants to determine the
characteristics of the customers most likely to purchase these products.
The supermarket chain has a customer loyalty program. As an initial buyer incentive plan, the
supermarket has provided coupons for these organic products to all of their loyalty program
participants and collected some data that includes whether these customers purchased any of
the organic products.
Consider the following econometric model:
buy∗i = β0 + β1 agei + β2 age
2
i + β3 tenurei + β4 metroi
+ β5 loyalty1i + β6 loyalty2i + β7 loyalty3i
+ β8 region1i + β9 region2i + β10 region3i + β11 region4i + εi (9)
where εi|Xi ∼ (0, σ2ε) and:
buy∗ = a latent variable that determines the decision to purchase organic products
age = Customer age, in years
tenure = length of time member of loyalty program
metro = 1 if customer lives in a metro area, 0 otherwise
regionj = 1 if customer lives in region j, 0 otherwise, where j = 1, 2, 3, 4, or 5
loyaltyj = 1 if customer has loyalty status j, 0 otherwise, where j = 1, 2, 3, or 4
The omitted categories for the dummy variables are region5, and loyalty4.
Suppose you have a dataset of 3,171 observations which, in addition to the required data, also
provides the following indicator variable:
buy = 1 if individual i puchases any organic products, 0 otherwise
This suggests the following econometric model:
buyi = β0 + β1 agei + β2 age
2
i + β3 tenurei + β4 metroi
+ β5 loyalty1i + β6 loyalty2i + β7 loyalty3i
+ β8 region1i + β9 region2i + β10 region3i + β11 region4i + εi (10)
where:
buyi =
{
1 if buy∗i ≥ 0
0 if buy∗i < 0
The parameters of model (10) were estimated as a Probit model and the results are presented
in Figure 10.
Note that the probability density function for a standard normal random variable (Z) is given
by:
φ(Z) =
1√
2 pi
exp
(−Z2
2
)
Page 15 of 23







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a) [4 marks] Let pˆi represent the predicted probability that an individual purchases organic
products, based upon their observed characteristics. Consider the following decision rule:
if pˆi ≥ 0.5 predict that b̂uyi = 1, otherwise b̂uy = 0.{
b̂uyi = 1 if pˆi ≥ 0.5
b̂uyi = 0 if pˆi < 0.5
Based on the information in Table 1, calculate the percentage of outcomes that are
correctly predicted? Using Table 1, comment on the usefulness of the model in predicting
buy = 1 or buy = 0.
true predicted frequency
0 0 2,141
1 0 650
0 1 128
1 1 252
3,171
Table 1: Predicted Probability Threshold pˆi ≥ 0.5
Page 16 of 23
b) [5 marks] Calculate the marginal effect for the variable age for an individual currently
50 years of age, with a tenure of 5 years in the loyalty program, currently not living in
a metro area, residing in region 5, and currently at the top tier of the loyalty program
(tier 4). Provide a clear interpretation of your calculated marginal effect.
c) [5 marks] Explain how you would calculate the average marginal effect (AME) for living
in a metro area for an individual currently 50 years of age, with a tenure of 5 years in the
loyalty program, residing in region 5, and currently at the top tier of the loyalty program
(tier 4). You have not been provided with enough information to actually calculate this
marginal effect so do not attempt to calculate the marginal effect. Instead, your answer
should clearly explain how you would calculate this marginal effect.
Page 17 of 23
Some Useful Formulas
Variance of the Sum of Two Random Variables
VAR(aX + b Y ) = a2 VAR(X) + b2 VAR(Y ) + 2 a b COV(X, Y )
Sample Variance
V̂AR(X) =
∑N
i=1 (xi − x¯)2
N − 1
Sample Covariance
ĈOV(X, Y ) =
∑N
i=1 (xi − x¯) (yi − y¯)
N − 1
Multiple Linear Regression Model
yi = β0 + β1X1i + β2X2i + . . . βK XKi + εi
OLS Residuals
êi = yi − (b0 + b1X1i + b2X2i + . . . bK XKi)
Estimator of Error Variance
σ̂2 =
∑
ê2i
N −K − 1 =
RSS
N −K − 1
Sample t Statistic
t =
bk − βk
se(bk)
Sample F-statistic
F =
RSSR −RSSUR/M
RSSUR/(N −K − 1) =
(R2UR −R2R)/M
(1−R2UR)/N −K − 1
when the dependent variable in both the restricted and unrestricted model is the same. Here
M denotes the number of restrictions, N denotes the sample size, and K + 1 the number of
estimated parameters in the unrestricted model.
Sample F-statistic for Test of Overall Significance
F =
(TSS −RSS)/K
RSS/(N −K − 1) =
R2/K
(1−R2)/(N −K − 1)
R2
R2 =
∑
(ŷi − y¯)2∑
(yi − y¯)2 = 1−
RSS
TSS
Adjusted R¯2
R¯2 = 1− RSS/(N −K − 1)
TSS/(N − 1)
Page 18 of 23
Probit Model
Latent Variable Formulation
y∗i = β0 + β1X1i + β2X2i + . . . βK XKi + εi εi|Xi ∼ N (0, σ2ε)
Response Probability
Pr(Yi = 1) = Φ
(
β0
σε
+
β1
σε
X1i +
β2
σε
X2i + . . .
βK
σε
XKi
)
where Φ(·) is the cumulative distribution function for the standard normal distribution.
Marginal Effect if Xj is a continuous variable
∂Pr(Yi = 1)
∂Xij
= φ
(
β0
σε
+
β1
σε
X1i +
β2
σε
X2i + . . .
βK
σε
XKi
)
βj
σε
where φ(·) is the probability density function for the standard normal distribution.
Marginal Effect if Xj is an indicator (dummy) variable
∂Pr(Yi = 1)
∂Xij
≈ Φ
(
β0
σε
+
β1
σε
X1i +
β2
σε
X2i + . . .
βj
σε
+ . . .
βK
σε
XKi
)
Xij=1
− Φ
(
β0
σε
+
β1
σε
X1i +
β2
σε
X2i + . . .
βK
σε
XKi
)
Xij=0
White Test for Heteroskedasticity
Econometric Model of Interest
yi = β0 + β1X1i + β2X2i + . . . βK XKi + εi
Auxiliary Regression
eˆ2i = γ0 + γ1 Z1i + γ2 Z2i + . . . γK ZMi + υi
The test statistic is N R2 ∼ χ2(M) where N is the sample size, R2 is the R2 from this auxiliary
regression, and (M + 1) is the number of parameters in the auxiliary regression.
LM Test for First Order AR(1) Autocorrelation
Econometric Model of Interest
yt = β0 + β1X1t + β2X2t + . . . βK XKt + εt
Auxiliary Regression
eˆt = γ0 + γ1X1t + γ2X2t + . . . γK XKt + ρ eˆt−1 + υt
The test statistic is T R2 ∼ χ2(1) where T is the sample size, and R2 is the R2 from this
auxiliary regression.
Page 19 of 23
Dickey-Fuller Test
• Version I: (no constant, no trend):
∆yt = γ yt−1 + υt
• Version II: (constant, no trend):
∆yt = α + γ yt−1 + υt
• Version III: (constant, trend):
∆yt = α + δ t+ γ yt−1 + υt
Augmented Dickey-Fuller test
With intercept :
∆yt = α + γ yt−1 +
m∑
s=1
δ∆ yt−s + υt
Page 20 of 23
C
ri
ti
ca
l
V
al
u
es
fo
r
th
e
5%
U
p
p
er
T
ai
l
P
ro
b
ab
il
it
ie
s
of
th
e
F
D
is
tr
ib
u
ti
on
Table 4: Critical Values for 5% Upper Tail Probabilities of the F Distribution
Page 6 of 9
Page 21 of 23
Right-Tail Critical Values for the t-distribution
Table 2: Critical Values for Upper Tail Probabilities of Student’s t distribution
Shaded Area = P (t ≥ tν,α) = α
For example,
P (t ≥ t3,0.05) = P (t ≥ 2.3534) = 0.05.
Note that ν denotes the degrees
of freedom of the distribution.
ν 0.1 0.05 0.025 0.01 0.005 ν 0.1 0.05 0.025 0.01 0.005
1 3.0777 6.3137 12.7062 31.8210 63.6559 28 1.3125 1.7011 2.0484 2.4671 2.7633
2 1.8856 2.9200 4.3027 6.9645 9.9250 29 1.3114 1.6991 2.0452 2.4620 2.7564
3 1.6377 2.3534 3.1824 4.5407 5.8408 30 1.3104 1.6973 2.0423 2.4573 2.7500
4 1.5332 2.1318 2.7765 3.7469 4.6041 31 1.3095 1.6955 2.0395 2.4528 2.7440
5 1.4759 2.0150 2.5706 3.3649 4.0321 32 1.3086 1.6939 2.0369 2.4487 2.7385
6 1.4398 1.9432 2.4469 3.1427 3.7074 33 1.3077 1.6924 2.0345 2.4448 2.7333
7 1.4149 1.8946 2.3646 2.9979 3.4995 34 1.3070 1.6909 2.0322 2.4411 2.7284
8 1.3968 1.8595 2.3060 2.8965 3.3554 35 1.3062 1.6896 2.0301 2.4377 2.7238
9 1.3830 1.8331 2.2622 2.8214 3.2498 36 1.3055 1.6883 2.0281 2.4345 2.7195
10 1.3722 1.8125 2.2281 2.7638 3.1693 37 1.3049 1.6871 2.0262 2.4314 2.7154
11 1.3634 1.7959 2.2010 2.7181 3.1058 38 1.3042 1.6860 2.0244 2.4286 2.7116
12 1.3562 1.7823 2.1788 2.6810 3.0545 39 1.3036 1.6849 2.0227 2.4258 2.7079
13 1.3502 1.7709 2.1604 2.6503 3.0123 40 1.3031 1.6839 2.0211 2.4233 2.7045
14 1.3450 1.7613 2.1448 2.6245 2.9768 45 1.3007 1.6794 2.0141 2.4121 2.6896
15 1.3406 1.7531 2.1315 2.6025 2.9467 50 1.2987 1.6759 2.0086 2.4033 2.6778
16 1.3368 1.7459 2.1199 2.5835 2.9208 60 1.2958 1.6706 2.0003 2.3901 2.6603
17 1.3334 1.7396 2.1098 2.5669 2.8982 70 1.2938 1.6669 1.9944 2.3808 2.6479
18 1.3304 1.7341 2.1009 2.5524 2.8784 80 1.2922 1.6641 1.9901 2.3739 2.6387
19 1.3277 1.7291 2.0930 2.5395 2.8609 90 1.2910 1.6620 1.9867 2.3685 2.6316
20 1.3253 1.7247 2.0860 2.5280 2.8453 100 1.2901 1.6602 1.9840 2.3642 2.6259
21 1.3232 1.7207 2.0796 2.5176 2.8314 120 1.2886 1.6576 1.9799 2.3578 2.6174
22 1.3212 1.7171 2.0739 2.5083 2.8188 140 1.2876 1.6558 1.9771 2.3533 2.6114
23 1.3195 1.7139 2.0687 2.4999 2.8073 160 1.2869 1.6544 1.9749 2.3499 2.6069
24 1.3178 1.7109 2.0639 2.4922 2.7970 180 1.2863 1.6534 1.9732 2.3472 2.6034
25 1.3163 1.7081 2.0595 2.4851 2.7874 200 1.2858 1.6525 1.9719 2.3451 2.6006
26 1.3150 1.7056 2.0555 2.4786 2.7787 ∞ 1.2816 1.6449 1.9600 2.3263 2.5758
27 1.3137 1.7033 2.0518 2.4727 2.7707
Page 3 of 9
Shaded Area = Pr(t > tυ,α) = α
For example:
Pr(t > t45,0.05) = Pr(t > 1.6794) = 0.05
Note: υ denotes the degrees of freedom of the distribu-
tion
Critical Values for Upper Tail Probabilities of the (Student’s) t-Distribution
df Upper Tail Probability df Upper Tail Probability
υ 0.100 0.050 0.025 0.010 0.005 υ 0.100 0.050 0.025 0.010 0.005
1 3.0777 6.3137 1 .7062 31.8210 63.6559 28 1 3125 1.7011 2.0484 2.4671 2.7633
2 1.8856 2.9200 4.3027 6.9645 9.9250 29 1.3114 1.6991 2.0452 2.4620 2.7564
3 1.6377 2.3534 3.1824 4.5407 5.8408 30 1.3104 1.6973 2.0423 2.4573 2.7500
4 1.5332 2.1318 2.7765 3.7469 4.6041 31 1.3095 1.6955 2.0395 2.4528 2.7440
5 1.4759 2.0150 2.5706 3.3649 4.0321 32 1.3086 1.6939 2.0369 2.4487 2.7385
6 1.4 98 1.9432 2.4469 3.1427 3.7074 33 1 3077 1.6924 2.0345 2.4448 2.7333
7 1.4149 1.8946 2.3646 2.9979 3.4995 34 1.3070 1.6909 2.0322 2.4411 2.7284
8 1.3968 1.8595 2.3060 2.8965 3.3554 35 1.3062 1.6896 2.0301 2.4377 2.7238
9 1.3830 1.8331 2.2622 2.8214 3.2498 36 1.3055 1.6883 2.0281 2.4345 2.7195
10 1.3722 1.8125 2.2281 .7638 3.1693 37 .30 9 1.6871 2.0262 2.4314 2.7154
11 1.3634 1.7959 .2010 2.71 1 3.1058 38 3042 1.6860 2.0244 2.4286 2.7116
12 1.3562 1.7823 2.1788 2.6810 3.0545 39 1.3036 1.6849 2.0227 2.4258 2.7079
13 1.3502 1.7709 2.1604 2.6503 3.0123 40 1.3031 1.6839 2.0211 2.4233 2.7045
14 1.3450 1.7613 2.1448 2.6245 2.9768 45 1.3007 1.6794 2.0141 2.4121 2.6896
15 1. 406 1.7531 2.1315 2.6025 2.9467 0 .2987 1.6759 2.0086 2.4033 2.6778
16 1.3368 1.7459 .1199 2.5835 2.9208 60 .2958 1.6 06 2.0003 2.3901 2.6603
17 1.3334 1.7396 2.1098 2.5669 2.8982 70 1.2938 1.6669 1.9944 2.3808 2.6479
18 1.3304 1.7341 2.1009 2.5524 2.8784 80 1.2922 1.6641 1.9901 2.3739 2.6387
19 1.3277 1.7291 2.0930 2.5395 2.8609 90 1.2910 1.6620 1.9867 2.3685 2.6316
20 1.3253 1.7247 2.0860 2.5280 2.8453 100 1.2901 1.6602 1.9840 2.3642 2.6259
21 1.3232 1.7207 2.0796 2.5176 2.8314 120 1.2886 1.6576 1.9799 2.3578 2.6174
22 1.3212 1.7171 2.0739 2.5083 2.8188 140 1.2876 1.6558 1.9771 2.3533 2.6114
23 1.3195 1.7139 2.0687 2.4999 2.8073 160 1.2869 1.6544 1.9749 2.3499 2.6069
24 1.3178 1.7109 2.0639 2.4922 2.7970 180 1.2863 1.6534 1.9732 2.3472 2.6034
25 1.3163 1.7081 2.0595 2.4851 2.7874 200 1.2858 1.6525 1.9719 2.3451 2.6006
26 1.3150 1.7056 2.0555 2.4786 2.7787 ∞ 1.2816 1.6449 1.9600 2.3263 2.5758
27 1.3137 1.7033 2.0518 2.4727 2.7707
Page 22 of 23
Critical Values for Upper Tail Probabilities of the χ2 Distribution
df Upper Tail Probability (α)
υ 0.2500 0.1000 0.0500 0.0250 0.0100 0.0050
1 1.3233 2.7055 3.8415 5.0239 6.6349 7.8794
2 2.7726 4.6052 5.9915 7.3778 9.2103 10.5966
3 4.1083 6.2514 7.8147 9.3484 11.3449 12.8382
4 5.3853 7.7794 9.4877 11.1433 13.2767 14.8603
5 6.6257 9.2364 11.0705 12.8325 15.0863 16.7496
6 7.8408 10.6446 12.5916 14.4494 16.8119 18.5476
7 9.0371 12.0170 14.0671 16.0128 18.4753 20.2777
8 10.2189 13.3616 15.5073 17.5345 20.0902 21.9550
9 11.3888 14.6837 16.9190 19.0228 21.6660 23.5894
10 12.5489 15.9872 18.3070 20.4832 23.2093 25.1882
11 13.7007 17.2750 19.6751 21.9200 24.7250 26.7568
12 14.8454 18.5493 21.0261 23.3367 26.2170 28.2995
13 15.9839 19.8119 22.3620 24.7356 27.6882 29.8195
14 17.1169 21.0641 23.6848 26.1189 29.1412 31.3193
15 18.2451 22.3071 24.9958 27.4884 30.5779 32.8013
16 19.3689 23.5418 26.2962 28.8454 31.9999 34.2672
17 20.4887 24.7690 27.5871 30.1910 33.4087 35.7185
18 21.6049 25.9894 28.8693 31.5264 34.8053 37.1565
19 22.7178 27.2036 30.1435 32.8523 36.1909 38.5823
20 23.8277 28.4120 31.4104 34.1696 37.5662 39.9968
25 29.3389 34.3816 37.6525 40.6465 44.3141 46.9279
30 34.7997 40.2560 43.7730 46.9792 50.8922 53.6720
35 40.2228 46.0588 49.8018 53.2033 57.3421 60.2748
40 45.6160 51.8051 55.7585 59.3417 63.6907 66.7660
45 50.9849 57.5053 61.6562 65.4102 69.9568 73.1661
50 56.3336 63.1671 67.5048 71.4202 76.1539 79.4900
60 66.9815 74.3970 79.0819 83.2977 88.3794 91.9517
80 88.1303 96.5782 101.8795 106.6286 112.3288 116.3211
100 109.1412 118.4980 124.3421 129.5612 135.8067 140.1695
120 130.0546 140.2326 146.5674 152.2114 158.9502 163.6482
END OF EXAMINATION
Page 23 of 23

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